I contacted the developer of the Obsidian git community plugin, who was amazing and replied really quickly. It turns out I just didn’t understand how to use git properly so I’m posting the answer in case anyone makes the same mistake.
I was running git status after renaming notes in Obsidian and getting concerned because it looks like git sees it as a file deletion and recreation. Turns out I should have had the confidence to run git add, which detects the rename. You can confirm this by running git status again afterwards.
git mv (which I’ve always used in the past) just renames and adds the file(s) to the staging area.
So I had no need to worry: git does work nicely with the Obsidian rename function. I just needed to use git add as normal to add the renamed files to the commit.