The Verdict: Unlikely
The researcher card fits your profile, but without a hosting agreement there is nothing to apply for yet. The agreement comes with a position, so the job search comes first.
The hosting agreement (convention d'accueil) is issued by an approved French research or higher education body: universities, the large public research organisations and many private institutes.
It comes with a postdoctoral contract, a research post, a funded visiting appointment or university teaching. When you are offered a position, ask the institution's international office to issue it; they do it routinely.
Once you have it, the process is quick: the decision is due within 60 days, and the card can run up to four years with your family included.