The ROMs
Updated and revised 11-29-20
Generally I disabled any stock stuff I could that I would not use. Anything past CM 11 has a maddening issue I call "peek-a-boo keyboard", most noticeable when using the browsers. The keyboard drops out while you are typing or simply refuses to open without a struggle. For CM 12.1, disabling the resident Android WebView app seems to fix the problem. For the rest, well it just gets worse as you ask more from the device.
Note: I have noticed that both my Nook Tablets (8 and 16 GB) run better if they've been on for a bit. If yours is cranky when it gets up in the morning, pouring coffee on it won't help, but a little siesta after boot may be just the thing
CM 11 



https://www.mediafire.com/?gq1y21uz11lr1c0
Flashed with SD-based TWRP 2.8.6.0, enabled signature spoofing with Tingle
This is arguably the best choice for the BNTV250a if you want something other than the stock B&N ROM. It runs smoothly enough that you can even keep the PlayStore app on it and it will still behave reasonably. It would be even better without it. The only drawback with this ROM is that Android Webview cannot be updated. This means that some websites will not display properly (XDA, for example!) or may simply crash, regardless of what browser you choose to use. I found the antique stock browser to be surprisingly good, but Via also works well. Also, the SSL is in need of updating, so you will run into issues with some websites and apps like OverDrive.
The lesson that this ROM taught me is that these tablets are for casual use. You can't just load them up with apps and expect them to run well. Think about what you really would like to use the tablet for. Consider older versions of apps that will probably still have more features than you will ever want but are less taxing to the system. You can see from the attached screenshots how I eventually pared my choices down. I should probably pare it down even more. Most of the apps are older versions and I've kept the total count low, certainly less than on my 16 GB tablet and other devices. You don't want to exceed about half of the internal storage capacity.
That said, I did find the Trebuchet launcher troublesome and replaced it with Nova Launcher. I also don't care for the stock CM file manager and replaced it with an old version of ES File Explorer I use on other devices.
CM 12.1 

http://www.mediafire.com/file/7zxc2e...K6-acclaim.zip
Flashed with SD-based TWRP 2.8.6.0, enabled signature spoofing with Tingle
Before you start installing apps the behavior is sprightly, even on the BNTV250a. As the internal space decreases and RAM use increases, it becomes a little less so. Even so, when running well I think it comes close to CM 11 (with GApps), albeit with more lag.
The stock browser seems like the best bet (I tried many), though it can be slow. It also tends to self-close when you start to download a file (the download continues and completes). The "peek-a-boo" keyboard with all browsers seems to be "solved" by disabling the stock Android WebView app.
For me this ROM has two major issues. When it is unhappy, it will simply shut down the tablet, literally turning it off. This used to sometimes happen when I ran it on my 16 GB model, so it's not the fault of the BNTV250a. The other issue is problems for apps writing to the external SD card. For a tablet with precious little internal space, you just don't want apps downloading into internal space or storing lots of data there. There is an edit to the file /system/etc/permissions/platform.xml which corrects this issue for some apps. Find the section of platform.xml below and add the last line:
Code:
<permission name=”android.permission.WRITE_EXTERNAL_STORAGE” >
<group gid=”sdcard_r” />
<group gid=”sdcard_rw” />
<group gid=”media_rw” />
This change enabled OverDrive and Kodi, for example, to download files to the external SD card. It might work for others.
Not as fluid as CM 11, but WebView issues are gone and you can access most up-to-date content (if slowly).
CM 13
https://archive.org/download/cmarchive_snapshots/cm-13.0-20160819-SNAPSHOT-ZNH5YAO0IN-acclaim.zip
Flashed with SD-based TWRP 2.8.6.0, enabled signature spoofing with Tingle
I was actually pleasantly surprised by its behavior on the BNTV250a--until I started to install apps. I had no problems with writing data or downloads to the external SD card. I kept the card formatted as "portable" because it was convenient for me, changing from ROM to ROM as I experimented, but the ability to extend the internal storage on to the external SD card is a real plus for a device with so little internal storage space.
If you elect portable storage you will eventually run into an app that can't write there. Kodi is an example, I'm sorry to say. I've tried many "solutions" to this problem but the only one that worked for me is to edit /data/system/packages.xml, adding the permission for each specific app you want. This is a long file and I did this using the editor from the CM File Manager. There are probably easier ways. After locating the app you want in the xml file, add this line anywhere between the <perms> and </perms> tags:
Code:
<item name="android.permission.WRITE_MEDIA_STORAGE" granted="true" flags="0" />
This change will be overwritten if the app updates, so you might want to keep that in mind.
The "peek-a-boo keyboard" issue is maddening (
not solved by disabling Android WebView) and many apps are so slow you really need to take your blood pressure medicine first. Not a good option unless maybe you install only one or two apps.
AOSP 6.01 "Marshmallow"
https://notredame.app.box.com/s/26a4bygh9vbaw7jjq08xr5evomvaw5ww/folder/7651670774
Flashed with emmc-based TWRP 3.0.2-0, enabled signature spoofing with Tingle
This is one of the ROMs built by
@amaces. The AOSP builds are very lean compared to CM ROMs so I was hoping this might perform somewhat better than CM 13. But I don't think it does.
The first big hurdle is enabling root. My thanks to
@digixmax for pointing out the right tool. After first boot (or maybe even before, but I did it after) you need to flash
UPDATE-SuperSU-v2.46.zip. Works perfectly.
The stock launcher is, well, funky. I replaced it with Nova Launcher but after installing a few other apps it was clear the system was under great stress. Lots of black screens and crashes. Not a good option for the BNTV250a.
AOSP 7.0 
https://www.mediafire.com/?5vcw8l603d3r5sc
Flashed with emmc-based TWRP 3.0.2-0, enabled signature spoofing with Xposed
This is another ROM by
@amaces and what I run on my 16 GB tablet. On that device it is virtually flawless (with GApps). I really wanted to say the same for the BNTV250a, but I can't.
Enabling root for the ROM is achieved by installing me.phh.superuser.1033 (attached below) after first boot. The ROM appears to be odexed to Tingle and so you must flash the correct Xposed (for 7.0) and install the installer app along with the FakeGapps app as described in the previous post.
The stock launcher is pitiful and you need to install a root file manager to get anything done. The system is laggy and
video is not smooth. Not even a potential option for the BNTV250a.
Lineage 14.1 
https://notredame.app.box.com/s/26a4bygh9vbaw7jjq08xr5evomvaw5ww/folder/3262390066
(there are many versions of this ROM. The link takes you to the "obsolete" folder where you need to hunt down
lnos_acclaim-ota-NJH47F.180211.zip. This seems to be the only version without the shut-down bug)
Flashed with emmc-based TWRP 3.0.2-0, enabled signature spoofing with Tingle
This is a third ROM built by
@amaces (his most recent for the NT). I didn't have much hope for this because if 7.0 was an issue, I didn't expect 7.1 to perform well.
The only native root capability is for ADB. To enable full root, you need to flash addonsu-14.1-arm.zip (attached below). I did that after first boot but perhaps it can be done along with the ROM.
Not surprisingly the ROM has the feel of CM. There appears to be a file manager but it is only a partial reflection of the native file explorer and is useless for the manipulations that need to be done, so you need to adb install a root file manager as described in the previous post.
The stock browser is complete rubbish. It looks to be Jello(?) but it is the most infuriating collection of electrons I have ever encountered. There are no navigational buttons, no bookmark button--no buttons at all! I really tried. I used gestures (some rude), I tried magic words (many of which do not appear in children's books). Nada. When I finally managed to reach my test site it was like wading through, well, Jello! So....attached below is the nice lightweight stock browser from AOSP 7.0. Navigate to /system/app/Browser and there you will find Browser2.apk. Rename it to something else and copy in the Browser2.apk attached below, setting file permissions to rwx-r-r. Reboot. If you are successful, you can delete the old, useless browser apk. To enable bookmarks, navigate to /system/app/BookmarkProvider and inside you will find the apk file. Tap to install. I don't know why this needs to be done, but it does even on AOSP 7.0. Reboot and you've got a functioning browser (rife with "peek-a-boo keyboard")
The ROM hates Kodi, crashing constantly.
Video from other apps is not smooth. This particular version seems to lack the shut-down bug that plagues nearly every ROM after CM 13, but it's a small comfort. Not a viable choice for the BNTV250a.