F-Droid unfortunately can’t build and deliver apps with Firebase in it, so we would need to implement another push service. If you can find a developer who wants to take up this task, then there’s a chance that may happen. Otherwise, there’s always the option to install the APK manually: https://karrot.world/app.apk
Edit: it might be worth investigating how Telegram does it, they have a version on F-Droid.
We could host our own F-Droid repo, like the Briar messaging app does. This would help getting updates if you’re not installing from Play Store.
It seems not super hard and doesn’t need much internal knowledge about Karrot, so could be something that a new contributor would want to work on. Needs a server, though.
Side note: Telegram on F-Droid is stripped of GCM/FCM and uses a fallback always-on push service.
May be worth checking if LineageOS or another custom ROM is available for your phone.
But I think there’s no reason why Karrot shouldn’t work on your phone. Can install it directly from https://karrot.world/app.apk? You may need to allow installs from APKs somewhere in the Android settings.