Do I really need to fill the tub with water to properly apply caulking?
While caulk is fairly elastic, it handles better under compression than tension. By filling the tub all the way, you expand the gap quite a bit. Once the caulk is applied and the tub drained, the caulk compresses.  During normal use, the caulk will likely never be in tension. As you'd have to add more weight than a tub full of water, before the caulk had to fill a gap larger than when it was applied.

Also, there's an old saying. "Happy wife, happy life" - Unknown