what ?! Think terminator 2 here ! He spends 3/4 of his ammo shooting from his motorcycle in an epic chase scene ! :p
That was a sawed-off Winchester Model 1887 with a pistol grip and modified lever action (so he could "flip-cock" it). And he was a full borg.
Theoretically, if you were a full borg or a troll, you could probably fire it from a motorcycle...with some penalties. I guess it would depend on how pink your GM's mohawk is.

In a low-realism, ride-the-blast-wave style campaign, go for it. In a more "actual physics" realm, the probability of horizontally parking your bike would approach 1.
If you wanted something to fire from a bike, I'd go with an SMG or a pistol. Long guns are, pretty much by definition, two-handed...shotguns are meant to be braced and in an aggressive firing stance, and sniper rifles are generally meant to be fired prone. From the back of a motorcycle - I'd pile on a lot of penalties to hit and/or keep your face off the pavement.
Anything less, and I would (if I were your GM) have you make some serious rolls to avoid a rather spectacular crash. A 12-gauge shotgun has...uh...a
fair bit of recoil. Like knock-you-on-your-ass-if-you-don't-brace-it-right recoil. Even a big guy, if he doesn't get that stock settled tight into his shoulder, is getting a nice reminder...in the form of a contusion. Your guy is BOD 3 / STR 3 - he's going to get knocked around a bit.
Then, if you go an chop the barrel and stock, you remove a bunch of recoil-absorbing mass...well, you'd be lucky not to break your wrist, and you're likely NOT going to control a bike very well when you're effectively getting kicked in the shoulder by a Awakened mule every time you pull the trigger.
If you really want a shotty, check out the Streetsweeper...it's a modified heavy pistol that is basically a shotgun (lower damage, implying higher gauge).
-Jn-
Ifriti Sophist