Paypal isn’t an issue because it’s no problem at all to convert the money in your account into euros and pay it into a French account.
After I moved I initially kept on a couple of UK clients who would only pay into a UK bank account, and I never worked out an answer. I didn’t stress over it because they were a very small proportion of income, but it’s one of the reasons that I let them go as soon as I could (the other reason being that I found it hard to compete with UK freelancers, what with charges here being so much higher and the rates I got from UK clients being so much lower than what French clients pay). If there had been more of them and I’d wanted to keep them, I suppose I would have set up a separate UK bank account as well as a separate French bank account.
But you definitely need to issue invoices for everything. Even if you never send them to the client, you still need them for your own records. If asked, you have to be in a position to provide a complete and properly numbered set of invoices with all the mentions légales on them, that exactly matches each transaction. Otherwise you are leaving yourself open to being fined big time for not keeping proper accounts.
The micro entrepreneur scheme doesn’t have a lot of rules but it is quite strict on invoicing procedures, to combat fraud. Those are the conditions you accept when you sign up so if you want to use this particular scheme, you will have to either find a way to organise your business so that you respect the rules, or else do it your own way and cross your fingers you never get a contrôle. To be fair it is very rare for MEs to be contrôle-d unless something gets flagged up.