There is no way to get under it without cutting a hole in my living room
ceiling. The opposite wall is another bathroom, with that bathroom's tub
sitting right there in the way, so that's not an option either.

Here's a dumb question: is *any* deflection acceptable for a second-floor
bathtub? I plan on selling this house in two years and the last thing I want
to do/can afford to do right now is to have to rip out an entire bathroom,
which is what this will take if I'm to take out the tub.

At this point I'm thinking about filling the tub with water and trying to
caulk it again and see if the filled tub trick will work.

You could cut a hole in the head or footwall, you could cut a
hole in the floor/ceiling underneath, you could cut a hole
in the outside skirt of the tub, or you could disconnect the
overflow drain and lever, (which assembly is probably plastic)
shove the drop-pipe sideways, and shove a hose down under
the pipe to squirt agregate or expanding foam through.

It just depends on what surface you think you'll have the best luck repairing.