After performing the bucket test I'm confident in saying that there's a leak somewhere. The bucket lost about 1/2" while the pool lost closer to 3/4". I had suspected an issue because when I turn on the pool filter after it has been off for a long time it takes a while to clear out the air. The return jets blow lots of bubbles and it takes the better part of a minute before all the air is gone. It's an older pool (1980's?) with a single bottom drain. I believe that bottom drain is piped to the skimmer, where there's a floating valve that prevents the skimmer from ingesting air when the water level is below the skimmer.

Would the next step possibly be to put a cork in the bottom drain and skimmer holes, and perform the bucket test again to fully confirm if this is a plumbing issue vs a shell issue?

Is it possible to fix if there is indeed a leak deep underground in the pipes? I see no signs of a leak anywhere on the surface of the yard.