What's the most important thing to tell a babysitter about bedtime?

The exact bedtime and the order of the routine. If your sitter knows when bed is and what needs to happen before that - bath, teeth, story - they have enough to work with.

Should I write down the bedtime routine for my babysitter?

Yes - a short written checklist is much easier to follow in the moment than trying to remember a verbal briefing. Leave it somewhere visible like the kitchen counter or fridge.

What if my child refuses to go to bed for the babysitter?

Tell your child before you leave that the sitter knows the rules and is in charge of bedtime tonight. Give your sitter permission to be firm and to contact you if they're struggling.