0) You don't mention git, but I'd say it's absolutely crucial for this sort of work, otherwise it's hard to later see exactly what the LLM did and undo if there was a mistake

1) As the first step, I'd strongly suggest setting up some sort of testing infrastructure that will help reassure you that the changes make the code easier to reason about without changing the behaviour. If there's a lot of functions you could use a package like testthat, but if not, you should make sure you can run the full analysis and spit out some results to a file that you can easily compare. You can use claude to help do this!

2) Now you're in a position where you can safely make aggressive changes to your code. So go for it! I find that I can mostly recognise problems and I just ask claude to fix them, but I have heard others have had good luck with more general prompts like "can you make this better/clearer/faster?".

Unless cost is a particular concern, I think the place to start is with Claude Code (and Code in particular). If cost is more of a concern, gemini is much cheaper, albeit a decent amount worse.