No, you don't have to give them your password for ballooning battery. You can short-circuit the discussion with the Apple rep by saying 'you can wipe it if you need to'.
Had mine replaced a couple months ago, they still wanted access. I'm pretty sure they gave a good reason,but I forgot. Probably just to run diagnostics. I made them an account. Still not ideal but I think they could just boot in single user mode and give themselves one if they wanted to.
If they still need your password to repair your computer, they really need to update their procedures.
Back then I just winced, gave them my password, and crossed my fingers (seems to have worked out), but these days I just couldn't do that. There's all kinds of financial and identity stuff in there.