Anything is possible but I am not sure there is a beefy enough small mower tranny out there...I could be wrong tho....My grandfather had an old Snapper mower with a 6 speed tranny but it was friction drive so I could see that was not gonna last long....what mower were you looking at?....
You know, the new Toros have the handle serve as speed control. The harder you push, the faster it goes, but i still coudn't see it moving a bike. About all you could do I guess is remove the blade and put a fly wheel on there, but that's only one speed.
You might cold look at the way a riding mower has it's gears/drive belt setup.
I have a self propeled mower and i stood on it (240lbs.) and it pulled me and the mower. Its only 6HP and the way its inganged is a belt, so if you change the pulleys it will go quicker. You will never know unless you go for it.