If I’m getting a battery replacement do I have to back up my phone

I would never take an iPhone in for any kind of service without having an up to date backup in hand. Apple never guarantees anyone’s data, so if anything unplanned for were to happen, any data loss would be entirely on you.

Really, your backup(s) should always be kept up to date anyway. What if you lose your iPhone, or it gets damaged, or stolen?

You should, yes. It likely won't be necessary to restore, but it's always a good idea.