I know the answer is probably “You can’t!”, but you never know:
I’m planning to install GrapheneOS or LineageOS and do a full deGoogle. The only thing I’m not looking forward to is saying goodbye to this fantastic app: https://play.google.com/store/apps/details?id=com.natureguides.birdguide
Would there be any way for me to reinstall it without using Google Play?
You could probably use something like an apk extractor to get the apk file, and then install it from that.
I think you can if you login to your google account with the Aurora store. You won’t be able to buy a new apps but I think you can redownload and update paid apps that method. I believe I read that or heard that somewhere we’ll see if I can find it
That’s correct, I have done it before.
You could try Aurora Store and login there with your Google Acc. Aurora is available on F-Droid.