What sort of visa are you going to France on?
The landlords want French payslips basically because those are paid into a French bank account, from which the landlord will take your rent payments each month. The US payslips do him no good because the money is over in the US.
Your tax situation is going to be very complicated, you know. If hubby will be 19 weeks in the US a year, he won't qualify at all for the FEIE - needs to be a bona fide resident outside the US for an entire calendar year or 12 consecutive months with no more than 30 days back in the US. France, on the other hand, considers you tax resident in France with 183 days a year physical presence in France. You definitely want to figure out the tax side of this arrangement before you go much further.