I've seen those things in Tokyo. It does look risky and I'm surprised there's not been an accident.
It seems like the clientele is 100% gaijin. Also everyone was wearing those stupid outfits, and so I wonder if you're allowed to decline it.
I've seen them get ripped apart on Japanese internet forums, although internet tends to bring out the worst in Jpn people so I ignore them.
I will say that these MarioCars seem to be extremely popular and everyone who's done it seems to rave about it.
So if you're willing to take some risks, I'd ignore the haters and just go for it.
Sorry, cannot help with any specific companies or routes. Isn't there one that does both Rainbow Bridge and Shibuya Crossing?