Humoring the guy for the moment:
Note that "a shotgun shell inside a cut-off barrel at the end of a stick" will not and should not do as much damage as a shotgun.
A weapon's barrel is required to
accelerate the projectile. This is why weapons which are meant to maximize power and accuracy generally have longer barrels--it maximizes the amount of accelerating force which is applied to the projectile. The instant a projectile leaves the barrel, the gases from the burning propellant can no longer accelerate it--they're simply venting into the air instead.
If a shotgun shell (or any other cartridge) is exploded outside a weapon, the bullets don't really go anywhere--they aren't accelerated significantly by the explosion. Now, that doesn't mean it's safe to hold in your hand--you still have a miniature grenade--but it's not going to do nearly as much actual damage as it would if used in the proper fashion.
So, when your nutball straps three "shotgun barrels" to the end of the stick, we're assuming that they're extremely short--probably barely long enough to hold the shotgun shells, plus an extra inch or so. That means that a shotgun shell fired from this device will deliver less energy than a shotgun shell delivered from an actual firearm. So it's not really a question of "one shotgun does 2d10 (or whatever), so if I strap three shotgun shells on a stick it should do 6d10!"
More problems. The guy presumably has to have some sort of firing-pin mechanism on each barrel. This could be a primitive "slapper" sort of thing, like a zip-gun; or he could concievably try to use the actual shotgun trigger mechanisms. This would reaquire a hell of a jury-rigged trigger system, though, and would make the weapon even more heavy. The "best" route would probably be to have the shotgun shells sitting in barrels that are "floating" on another tube, such that when the barrels are swung into a target they're driven backward--driving the backs of the shotgun shells onto a fixed spike or needle.
Unfortunately, this is still an incredibly unreliable and shoddy firing mechansim. The potential for having some (or all) of the shells fail to fire is pretty high. Moreover, you have the problem of (a) getting all three shells to fire at once, and (b) making sure they fire at the right time, or don't fire accidentally. With the backwards-plunger system described above, you could blow off your foot just by dropping the thing, or falling, or accidentally smacking the thing into a wall or floor. Also, while it's possible to fire two shells simultaneously with a precision, well-manufactured double-barrel shotgun, doing so with a jury-rigged piece of




is a more dubious concept. And if one shell were to go off earlier than the others, it would at the very least reduce the damage of the weapon (because the head of the weapon, once one shell goes off, is going to be flying
away from the target at high velocity), and, at the worst, could cause the head to rotate or twist, causing the subsequent shells to go off at a suboptimal angle (and one which could injure the user).
So...uh...this thing would require a high degree of skill and luck to manufacture such that it didn't completely explode or fail on use. It would require a high degree of care and luck to carry and weild without injuring onesself, even outside of combat. And it would require an extraordinary degree of luck to use it successfully in combat.
That being said, the guy is obviously a pain, which I realize is a problem since he's so close to the rest of your group. You can try to dissuade him by passive-aggressive punishment--never letting him get into combat and boring him to tears, or constantly killing his characters because of the idiot things they do--but that's really just an attempt to drive him off instead of having to kick him out, isn't it?
It seems like your only real options--as far as this particular campaign is concerned--are to keep stringing him along while "punishing" him, in hopes that he quits, or kicking him out outright. Neither of which would be very easy on the rest of the party, I'm sure. I don't envy you.