You really shouldn't be wrangling a mini sentry unless the enemy is outside your range. Even then I can't really recommend it all that much. DPS is too low.
It beats the hell out of a pistol. =p
I like wrangling it -alot-.
And is in turn beaten mercilessly by "Place the sentry and shoot your shotgun or pistol alongside it."
Derp.
Not really. I will kill anyone way faster by wrangled mini sentry than mini+shot. And if you wanna engie duel, I will kill you way faster as well. Not only it does more damage, but it snaps at you, not wasting any dps.
Wrangled minisentry does 96dps.
A minisentry + shotgun does 81dps on average.
Don't get me wrong, if someone gets near melee range, I will drop wrangle and shotgun on the face which is where the shotgun shines, but at normal ranges, wrangle minisentry wins. Not only that it does more damage, but you can use its shield for cover.
Besides shotgun on the face at melee range, the next dps step for the engie is the normal sentry at level 2. If you really want to use your shotgun along with your sentry use the normal sentry at level 1 + the shotgun, then you will match the wrangled minisentry dps (it's almost exactly the same dps) and even have a longer range for the sentry, but you will have a longer setup time as well.
Back before the update, I used engie traps + shotgun, which was basically using a level 1 around a corner and luring enemies in for the kill.
Course, if you like doing mini & shotgun, nothing wrong with that, just don't come saying it does more dps because it doesn't. The pistol + mini will do more dps (around 10 more dps than wrangled sentry), but that's assuming you will not miss a single shot and the guy will stay in the mini sentry's range.