It's possible. I don't know how to program a flash ROM while it's in the circuit, but it couldn't be all that hard. If you have a seperate device to program the flash ROM, it's probably easier though.
With the existing multi-carts, they might've hacked the games to make them work on their mapper. (usually you see non-mapper and CNROM games on those multicarts because they need little to no modification to work on another mapper). And the type of modification also depends on what order the games are in on the ROM.
I've never seen a multicart with all MMC1 or MMC3 games before, maybe there's one out there though. Or maybe there's a good way to do it. Maybe using a latch or dipswitch or something on the highest address lines of the ROM would work, the highest address lines of course would be out of the range that the MMC1 normally connects to.