Looking at the manual, the force feedback wasn't just a simple rumble motor.
It shifted the front end of the bike on a slide left and right.
It has a separate drive board like the driving games.
I vaguely remember playing it long ago and think it worked like this:
when you lean into a right hand turn, the front of the bike will slide left.
It's counterintuitive to those who don't ride, but this is actually how it works on a real motorcycle.
If you get up to coasting speed on a bicycle and really pay attention to the handling, it does the same thing.
I don't remember any shaking, but I can't even remember how long ago I played it.
This is the reason I think the newer motorcycle games like fast and furious suck so bad.
You're just tipping the bike left and right. It doesn't feel anything like you're riding a motorcycle.
I really don't think there's much of a chance you'll get it to work without some type of custom programming.
Just hooking it up to a force feedback wheel is not going to give you the same effect.
In fact the effects would probably seem completely wrong.
If you know someone who is good with arduino and automation, you could rig it up so that the sliding action is just controlled by the handlebar movement (independent of the game).
That's beyond my abilities though.
There's also the option of bolting in a rumble motor from something else.
Doing both would be ideal, but it would be for the hardcore tinkerer.
EDIT: I'm gonna add that to steer on a motorcycle you don't turn the handlebars in the direction you want to go like a steering wheel.
You lean and the bars actually turn slightly in the opposite direction. You apply forward pressure to the grip on the side you want to turn, the bike leans over because the wheel is turned slightly in the opposite direction, and then the bike goes in the direction of the lean. Again, a bicycle drifting at a decent click works the same way (but at slow speed you turn the handlebars in the direction you want to go). You just don't realize it because it comes naturally.
I don't remember if ManxTT worked this way, but I do remember it was way more like riding a real motorcycle than any other arcade game I've played.