Thanks everyone for this chain of posts. Just thought I would add my experience as well in case it helps anyone. We are in a not too dissimilar situation with me being non-EU and my wife being French. I lived in France a while ago but moved to Switzerland on a B permit for work and did not renew my Carte de Sejour (CdS). As my wife recently found a job in Basel, we decided to live across the border and, as per Mepriana
's advice, I went to the mairie in St. Louis to get more info. The process turned out to be quite simple; the lady there gave me a list of documents (salary slips, proof of address in St. Louis, medical insurance, etc.) to bring to renew my CdS so I went back with all the right documents a few days later and I now have my temporary CdS! And it has only been two weeks or so! Probably because the mayor of St. Louis has special rights or something like that...and also because I used to live in France.
In the meantime, my company has been sorting out my G permit and my wife has received her permit through her company. Amazing how fast things move when they do move...partially makes up for all the days spent queuing at various prefectures and embassies!!!