[stumped]

bite the bullet and do the factory reset
amazon will not help you because the issues are with sideloaded books, and the way the fire manages and stores indexing is well hidden

things you could try first, that may help is
1.get to the list of (system) apps via settings, and delete all the cache and data you can find for anything to do with the books' apps
2. there is a soft reset option- less destructive than factory reset. hold down power button and ignore all messages until tablet restarts ( 30 - 40 secs). that will not wipe any books or apps.

with the Fire HD ( which is no longer called Kindle fire BTW) stuff like kindle reader, prime video etc are apps, albeit pre-installed ones that you can''t delete. so you can do "app" stuff like clear cache, data. ( that's a known fix e.g. for issues with prime video)

you other option, if you don't want to factory reset, and you mostly sideload, is to drop the whole Kindle thing. install google services, convert all your books to epub, then use an epub reader of your choice - Moon reader is good. That give you far more control.

you don't say what generation fire HD but anything running Fire OS 5 will allow you to install the 4 apps needed for google play store to work, and instructions are in other threads.

a slight concern is that you mention the carousel. that is a feature of OS4, i.e. the ~2013 devices, and there is no easy way to add google stuff to those.
check amazon / wikipedia & work out exactly what device you have, and look in settings to see what version of Fire OS is running. and chack that you have all available system updates.

to help with troubleshooting, explain exactly how you do the sideload from calibre steps