You can use
nxdumptool (the rewrite release) to do full cart backups with a modded switch. It dumps the card's contents (XCI), header, certificate, initial data, cardIDset, the HashFS partition and some other information as well. On a modded switch, the XCI is all you need to run a game. I think the MIG needs some of the additional data to "emulate" a valid game cart.
This is as good as it is going to get. With the data obtained by the tool you ought to be able to create a legit image for the MIG. Basically a clone of your existing cart. Granted, I am not familiar with what sort of data an XCI on a MIG needs to be able to run games on an unmodified console. I hope it is not some sort of proprietary format where only games from their dumper will work or something.
Either way, nxdumptool dumps pretty much all you can dump from a game cart, and if that data is not sufficient to create legit XCI images of your carts then you may have to go and buy the dumper.
Does the MIG come with any sort of documentation on how to piece together valid XCI files with the data output by nxdumptool? Considering you have the carts certificate, header, initial data, cardIDset and the hashFS partition you
should, in theory be able to utilize it with a MIG. Maybe someone in the community has done the process before and there's a tutorial or tool out there somewhere on how to create legit XCI files
not using their proprietary dumper.