I spent quite a bit of time trying to get LineageOS working with my Fire TV Stick Lite Remote, so I want to describe my problems and how I solved them to help others with similar issues.
I followed the instructions, boot to TWRP (I had TWRP and magisk installed), wiped data, system, and cache, and flashed lineage-18.1-20220505-UNOFFICIAL-sheldon.zip and open_gapps-arm-11.0-tvmini-20220215.zip.
Everything went smoothly until it's time to pair the remote. The screen would show the image of the remote, and the animation indicates that I need to push the back and home buttons. As soon as I clicked on the home button the countdown was replaced by a spinning circle, but after a while the pairing will fail and the countdown timer start again. I tried holding the Home button for 10+ seconds, tapping it for 10+ seconds, holding Home + Back for 10+ seconds, etc. All sorts of combinations. All failed. I gave up and pressed ESC on the USB keyboard attached to the Y-splitter and continue to the rest of the setup. Once the setup is finished, I went into setting, Accessories and Remotes, and tried pairing again. LineageOS will find the "AR" device as soon as I click on the home button, but pairing always fails ("cannot pair") in the same fashion as during the initial setup. I did notice that if I hold Back+Home then the pairing failed quickly rather than a long timeout.
I though that maybe something was wrong with the remote (came with my brand new stick, with fresh Amazon batteries). Measured the battery voltages and they were fine. Also some people say that the LED light on the remote should flash if Home button is held for 10+ seconds but mine does not. So I tried pairing the remote with an Android phone. The pairing was very easy. I only needed to press Home once to let Android detect the remote, then hold Back+Home for the pairing to be done. I can even use the remote to navigate around on the Android phone.
So the hardware is fine, and I know the stick's Bluetooth is fine too because I can pair it with a generic Bluetooth keyboard with no problem whatsoever. So I started googling and found that the problem may be cause by GApps:
https://xdaforums.com/t/dev-rom-uno...11-for-raspberry-pi-4-b.4212945/post-86089723
So I started from scratch, re-installed LineageOS but this time WITHOUT GApps. During the initial setup, things are even worse than before because the remote cannot even be detected. So I pressed ESC on the keyboard and got into the Home Screen, and proceed to Settings, Accessories and Remote to try to pair again.
Same problem, no device detected. So I tried to pair the remote with my Android Phone again. This time Android phone cannot detected the remote either. I took out the battery of the remote, turned Bluetooth off and the on on the phone, and finally I was able to pair the remote to the phone again.
Now I went back to try to pair the remote with the Fire TV Stick. This time it finally worked. The remote was detected as "AR" and I held Back+Home and then clicked on "pair" and finally the pairing was successful.
I also rebooted the stick a few time and the pairing continued to work.
I decided that I actually like the system without GApps. I installed F-Droid and then installed VLC from it. For everything else I just use adb install. BTW, if you are on Windows and all you need is adb, you don't need to install the whole Android SDK, you can just install this:
https://xdaforums.com/t/tool-minimal-adb-and-fastboot-2-9-18.2317790/