Can you site sources that say your husband can not legally be paid in the US? I'm not convinced this is true.
Since his employer is a foreign entity he will be responsible for both employer and employee contributions. He doesn't need to go through a payroll firm.
You need to talk to the commune to confirm his permit allows it and then talk to the AVS office to register and set up to pay those contributions.
The 'one client' issue does not come into play here as the employer is a foreign entity and can not be made to pay the employer contributions. This is only relevant for Swiss companies with Swiss freelancers.
Let's repeat: the employer is NOT Swiss based. The clue is in 'US based'.