Sidewinders and eagles are the smallest ships that are built with FTL travel in mind. Sidewinders probably sacrifice a lot of capabilities just so they could house an FSD. You could theoretically stuff a single sidewinder into the fighter bay, but it would be kinda pointless because fighters need a mothership to travel, sidewinders don't.
You'll also be limited to a single sidewinder, compared to having 8-16 fighters. I think that fighters are modular and are stored on the mothership in compact pieces. That is why it takes time to assemble another fighter every time you lose one. Sidewinders aren't built for that. They'd have to redesign the whole ship, but I don't think it would be a sidewinder anymore.