As an employee, ie portage salarial, you pay higher social contributions and in return you get better benefits, I think the main ones are unemployment entitlement and a higher rate of sick pay, I guess holiday pay too, and maybe other things.
If you set up as a micro entreprise, your 'patrimoine' ie your house is automatically protected. This has been the case for 2 or 3 years I think. So for instance a client can't sue you and you have to sell your house to pay. But if you're in a sector where sueing is the norm, obviously you should take out professional insurance for peace of mind.
Under PS you can't invoice the client, it's not possible. Your employment contract is with the PS company. They generate your payslip, pay your social contributions, and put the net salary into your bank. Invoicing the client would be like if you worked for an employment agency, say Reed, and Reed sent you on an assignment at say Sainsbury's. You get paid by Reed for the work you do at Sainsbury's. You can't decide you'd like to send your own invoice to Sainsbury's, because you are not on Sainsbury's payroll, Sainsbury's can't pay your social contributions, they are not your employer, you are employed by Reed. This makes life easier for you - you receive your salary at the end of the month regardless, and you don't have to worry about whether or not the end clients have settled up yet or not. That's part of what you're paying the PS company to take care of. But you would need to see if they could handle your business model - from what you've said, it's not too clear at what stage the contract would be drawn up and signed with each separate client.
A lot of French seem to think that setting up a UK Ltd Co is the answer to everything, because the UK has such a reputation for being a low-cost, easy country to do business in. Which compared to France I guess it is. But that route can be fraught with various pitfalls, you have two countries' tax systems to negotiate and you still have all the same decisions to make on the French side, so if you're looking for a simple straightforward option I really don't think that's it. You might end up paying more overall if you get it wrong, and if you get it "right" and succeed in paying less in France, there is the risk that France will claim the only reason you set your business up that way was to avoid paying your dues - which would be "mauvaise foi" - and if you couldn't demonstrate that there were business advantages over and above avoiding charges, they could impose penalties.