Best way to get proof of address is to get home insurance. You need it anyway before you move in, but you can put it on any credit card from anywhere. If you've already done one international move, you already know about the dumb circle of despair that exists everywhere of 'I need an address for a bank, but a bank for a phone number, but a residence permit for a phone, but an address for a permit' or whatever the variation is. The house insurance cuts the loop.
HSBC advice: if you phone their help line, it gives you a French menu. If you repeatedly say 'English' into it, even though it's a robot, it takes you to their UK call centre. It took me months to find that, and the 'English' helpline advertised on the website has a helpful welcome message in French. HSBC are pretty shit with customer service generally, although asking around it seems like all French banks are terrible compared to other countries. I've heard reasonable things about ING, but they're Dutch and online only in France.
Also, all French mobile and internet providers suck. Prepay is a scam and they steal your money and cancel the number almost instantly, but it's the fastest way to get a burner phone, before you break the circle of despair. Don't try to work from home for a few weeks, it takes forever to get internet connected, and Paris has lousy phone signal and barely any 4G coverage.
If you're working in any sort of public sector job, don't expect to get paid for about 3 months. Make sure you've got some savings, or use the time for an extreme diet. Don't expect HR to speak English unless you're working somewhere amazing (I love the suggestion of talking to HR! Mine shrugged and said in French 'not our problem' when I asked how I was meant to eat after they failed to pay me yet again.)
Can you tell I found the first few months tough?! The baguettes and wine nearly make up for it, although following the bureaucratic nightmares and the handling of the pandemic I'm looking for a job somewhere else (but definitely not the UK!)