Same situation as you but I just got past that application not installed thing on AccountAndSyncSettings.apk. Sort of a round about way to do it but I've now got my Fire configured with TWRP where I can flip back and forth between stock and CM7 easily with all my root tools on both so I'm doing a LOT of trial and error lol.
In rooted stock, I used Titanium Backup to backup all 4 apks and send the backups to dropbox. I sort of stumbled across the fact that I saw these backups when booted under CM7. At first I tried restoring all 4 apks with Titanium but that got me a whole bunch of force closing when I did anything with Amazon Video. I figured ok why not just restore the one that won't install then install the other 3 legit? Did that and it lets me get past the Register Your Kindle screen and on the Amazon website when I drill down in my account and look at my devices, I see an additional Kindle there now.
Unfortunately when I try and play anything, even if I temp-unroot with OTA first, I get the "Your device isn't configured..." message. I am going to try and drop in the DRMProvider.apk from stock in the same way I got the AccountsAndSyncSettings and see if that changes anything... It seems like I'm most of the way there now but hitting some DRM issue.
Update 11:52am (who needs sleep?!)
Dropping in the DRMProvider didn't work - same result. I started poking around on the file system and came across /data/system/accounts.db and in looking at that file on stock vs on CM7, I see there are various keys for video on demand, reading, MP3 and the Amazon App Store. I'm now wondering if pulling some of the data from the stock accounts.db into the CM7 accounts.db wouldn't get this working. Just dropping the stock file over top of the CM7 file after getting the 4 APKs installed doesn't work. Results in either a lot of force closes or, best case, you can get into Amazon Video, see the first screen but trying to drill down or click on anything results in a message along the lines of can't open this web page, try again later.
I bought the SQL Lite editor off Marketplace to mess around with this and it works fine on stock...and force closes instantly on CM7. Doesn't mean I can't merge the account.db data, just means I have to do it on the PC or on stock which slows things down. Grrr.