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.