Since you won't be in France, you won't be working in France. You therefore do not need permission to work in France.
Furthermore, as a contractor, the business relationship is between your corporation and the French company. That your corporation has a sole employee does not change the fact that you are employed by that corporation rather than the French company.
However, there is a caveat: all employment law that I am aware of defines "employee" so as to prevent employers from escaping legal requirements simply by paying their employees as contractors. In other words, the legal arrangement you describe is typically not sufficient to establish that you aren't an employee; other factors usually come into play such as the degree to which the client directs the working hours and conditions of the contractor. (In saying this, I don't have much specific knowledge of the relevant French law, but I do understand that it is similar to the law in other countries with which I am more familiar.)
In your case, the fact that you and your corporation are foreign probably strengthens the argument that you truly are a contractor. The fact that your personal income is received as an employee of the corporation may also have the same effect. Furthermore, modern employment law does not deal very well with remote cross-border employees. We occasionally get questions from people who want to work remotely for a foreign employer as an employee, and frankly that just doesn't seem to be possible: what employment law would apply? Which country's payroll tax system would apply? There just isn't a framework for this kind of relationship. The best way to do it seems in fact to be the arrangement you already have in place.
To reiterate, permission to work in France is only required if you're actually physically present in France when you do the work. Since you will not be, you do not require a French work permit.