I actually had a basement sewer backup some years ago, due to heavy rain - and yuck! Happily, my basement has tiled walls, so no drywall replacement to deal with.
But we did/do have basic indoor/outdoor carpet down there, and we had someone come and suck up the water and treat the carpeting. You only have to pull it up if there's padding underneath. They cleaned it and treated it with an enzyme product, and it has never smelled at ALL.
Anyway, in your situation, I'd buy some rugs and toss them down. The outdoor rugs for patios are all on clearance at this point, too.