This is just a wild guess, but it might be close. I've picked up old (around
1989-91) issues of Nintendo Power in various thrift stores, and more than once,
Nintendo World Championships or something like that have been mentioned. The
cart you're referring to may have been one of the carts used in the
championship.
Ok, I just found the March/April 1990 issue of Nintendo Power. It says that a
"triathalon" Game Pak will be used in the NWC (I was right, it was called
Nintendo World Championship) contests. The specially designed Game Pak runs for
about seven minutes and contains segments of three Nintendo games: Super Mario
Bros., Rad Racer and Tetris.
The game would probably be quite rare, considering that the cartridge was only
used during a contest. To my knowledge, that game was not released to the
public. Only eleven cities are mentioned in the list of tour stops, but there
was probably more than one copy of the game at each one. Still, it sounds very
rare.
I have no idea about the switches, though.