Pokemon Psuedo-Rival Run Help

Heya, I've been wanting to do something for a long while now, the idea is that 2 people play the same Pokemon game (Or opposing versions) and after every gym or 2, they have a battle and see whose team is better. Simple enough on real hardware, but I can't seem to find anyway to do so, connect via wireless connection and such. The only solution I can think of is Pokemon Showdown, but neither of us really want to rebuild out teams every time we battle. Anyone have any ideas about how I might go about this? Really any main series game would work, though GBA or later would be preferable.

Very short answer here, because I think you can figure it out. This is (as far as I'm aware) a solution for Gen 1-3; though it may just be 3 depending on the link capability.
You can use VBA-Link: http://www.vbalink.info/

There's also Desmume Wi-Fi (hosted on EP) or MelonDS: http://melonds.kuribo64.net/ ; MelonDS only works if the two players are both on 64-bit OSs. As I am currently not, I cannot confirm one way or the other.

Have fun.

Desmume Wi-Fi seems to be no longer up and running, unless I'm really misreading this somehow. And MelonDS might work but all the BIOS I can find have the wrong files, it's looking for .bin but 2/3 of my files are .rom files. No idea why, can't find anyone else with the same issue, also are you sure MelonDS's connection is online, I saw a few posts about it not being implemented yet, but I wasn't sure if they were for prior versions.

Then I would suggest VBA-Link.