Have you checked that your US licences are actually exchangeable as many states licences are not

Skip the French Driving Tests? Some Americans Can. -

If your U.S. state has a reciprocal agreement with , you can skip the French driving test and exchange your U.S. driver's license for a French one.

barefootblogger.com

The article gives a reasonable resume of what you should do.
One point to note is that you should provide a certificate from the state stating your test date etc which may be easier to get when you return to the US

 

Maybe hold off on exchanging your license until the visa issue is squared away.

 

for your year first year here use your US licenses ....
when you return on a VLS-TS do the exchange (if your state is one of those that permit license exchange) within the first 12 months of your arrival ... the sooner you do that the better to avoid any delays ... all you need is to validate the VLS-TS and have a proper address.

 

I don’t have a decisive answer, but it looks like you can obtain a temporary residence card.

If you work through the questions here, the only exemption to the one year deadline is for students. It may be that as short stay visa holders, you would not be deemed to have your “normal residence,” but I would not count on that.

 

the OP has a VLST not a VLS-TS hence they do not qualify for having the residence card, their visa allows them to stay for 12 months then they have to leave hence will not qualify as a valid residency for license exchange