Never even knew such a device existed before this.
First you and your friends would be considerably better served downloading a PC multiplayer version and getting USB to SNES adapters. If only because such a thing would handle latency and the like. For the other games then you are probably still better off finding something with Kaillera, or equivalent, support.
I would have guessed the xband was some kind of game genie equivalent (ROM patcher, possibly also some RAM patching options) and modem combo. This would tend to mean extra hardware that has to be emulated or replicated.
However the thought exercise is an interesting one and http://www.gamasutra.com/view/feature/131781/the_internet_sucks_or_what_i_.php
will probably be as relevant as ever. The games picked seem to have already been multiplayer games at their heart, doom being the main exception though that is at best a minor distinction and possibly one that helps here (latency is more likely to be easier to deal with).
I do not know what level of onboard storage the xband had (I have not even looked to see if someone has pulled one apart at this point), the idea of online services going off would have been something of a new one so I doubt people made the effort to save it. If it was some kind of downloadable patch held in RAM then there is very little chance.
Being two player I imagine there would be something of a host-client model. In Doom's case I would have made a code to disable all enemies, unlock all doors and maybe regenerate pickups after a certain time. I would have then had the modem send pickup takings both ways, player locations both ways, player movement directions if movement alone would not have been enough to handle sprite directions, firing info and then had the local games manage ammo counts and whatever else (a firing offence for a modern game, here though there were rather limited resources for proper anti cheat). Frag counts was a nice touch, I wonder what was done for that.
The rest of the xband would then have been the in game menu (by the looks of things a level select, one that probably ran "if the same then choose that, otherwise random" if one of those videos was anything to go by) and the backend at xband HQ (I am guessing a modem bank and the charging/score keeping architecture). For this project I would probably start out replicating backend using some kind of null modem and sorting out the rest via other channels ("send me your IP address...."), or just remaking the patch* and using Kaillera.
*do the unlocks, regenerate pickups... have the second controller control an "enemy" modded to be the doom marine and then send data around for what I imagine is some kind of viewport rendering (plus ammo counts) on the client connection.
This is an awful lot of effort for Doom.