In order to get onto the French public health system, you must be working in France. I'm pregnant too and just moved to France a few months ago. Although my husband is French, I'm not eligible for the public system since I'm not working here - just finishing up contract work in Germany. He's also working in Switzerland and was able to take out French private insurance. He could have opted for the public system but once you get on the public system in France, you can never go back to private.
We got insured with Credit Mutuel. There were some problems at first accepting my pregnancy (partly because I was 7 months) but we put pressure on them and they caved. My insurance costs about 200 euros a month and covers everything. Plus, being privately insured means I can see private doctors - was able to get all my ultrasounds after 6pm so my hubby could be with me. Private ultrasounds cost 200 euros and all were paid for no question. Mid-wife classes and private consultations also all paid for as well as postpartum pelvic re-education classes and treatments.
So I recommend checking in with Credit Mutuel - especially since you're still early in the pregnancy. They also have a special frontalier plan where you can divide a Swiss salary into a francs account and a euros account and also freeze the current Swiss currency rate for 6 months to a year. Feel free to PM me if you'd like more help!