Catalina Notes.app draining battery?
All seems to be resolved by toggling off and then back on the iCloud sync for Notes (In Catalina, this is found in the Apple ID section of System Preferences):




Turn off Notes iCloud sync
Turn it back on
Open Notes.app
Be momentarily terrified because all your notes have vanished
Watch as they trickle in from iCloud


Running Activity Monitor now lists Notes as barely having any energy impact whatsoever. Hooray!



BONUS: Back Up Your Notes

I was reluctant to try this method because I was terrified that I would introduce some sync problem and wipe out my notes. As Apple provides no easy way to make a local backup/export of your notes, I felt that this was a reasonable worry. 

Eventually, I found this little utility that makes .txt files (with HTML inside) of all of your notes (WARNING: it omits any images or attached files). It ain't great, but it worked and helped to put me at ease.

Apple also provides a way of downloading your iCloud data on their Data and Privacy site… which I initiated, but have yet to receive the download link. According to Apple, this could take up to 7 days.