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 try Aurora Store and login there with your Google Acc. Aurora is available on F-Droid.