Can you siphon bath water into toilet, remove side panel of bath, take pipe out, push ball through with stick, replace pipe, replace side panel, party time?
Siphoning the bath water out is going to be pretty tricky unless the bath is set pretty high up, though?
Have you tried something like a bath plunger? If you turn the rubber cup inside out (or flatten it as much as is possible) before placing it you’ll hopefully create a vacuum rather than plunging it further down.
Edit, sorry, @markg was posting as I was, and explained it better