Can a French national becoming a US resident legally keep their life insurance? Is there any legal restrictions, other than having to inform the bank and the IRS (because of FATCA)?
In the US, there's an excise tax on foreign insurance policies. The tax rates depend on the kind of insurance, for life insurance it's 1% of the premium.
You can find more information in the IRS publication 510.
I'd personally suggest avoiding this. There are plenty of insurance companies in the US.
Talk to a tax adviser (US-licensed EA/CPA) for more details and a professional advice.