I know if you go back to the character select screen, you can launch a new character set, and almost everything works the same as if launching the new set from the desktop.

I believe, the window layout is not changed, if the teams are different. Everything else should update fine.

I actually go to the desktop, just about every time that I change teams. But there hasn't been a queue to log in, in quite some time, so that's not really a penalty for me.

You could make use of virtual mapped keys. Something like, you push 1, which is your DPS key, and is virtualized as assorted dps options depending on the toon. I've not really done this at all; just have dps things on 1-3, heals on 7-9 etc, across any team.

But I don't really mix and match toons.

I have occasionally traded one toon from Team A for another toon on Team F. Then you have the drag the toon into the slot, drag the other toon out of the slot. Edit a few mapped keys, especially the round-robins or if a given team has a Keymap specific to the team.