Minecraft “game already running”

You actually can use a batch file to change %AppData%. In a batch file, changing %AppData% is not global; it only applies to the commands in that batch file. (It would be a massive security hole otherwise!) So if you want to go that route and use a batch file, you seem to already know how to do that and you don't have to let worries about global environment changes stop you. For multiple instances simultaneously, use multiple batch files with different %AppData% settings.

I use this method to keep my and my kid's Minecraft installs completely separate, so that our profiles, worlds, and even jar versions are inaccessible from the other's launcher.

Alternatively, if using the new launcher is an option, this is easily done within the profile settings. I detail the method in this answer on a question about keeping different game versions' worlds separate. That's exactly the method I use myself for keeping multiple different Forge-based modpacks separate.

The two methods can be combined for even finer control over world separation, if that is relevant to your use-case.

You need to use the Mojang account's email address as your username instead of your original username.

The reason it may have still been working after migration is proberbly because your existing account credentials/session data.