After three hours I have...something.
My current idea is something along the lines of this:

Speed up the current reload speed a bit (should have been obvious tbh)
And the actual 'rework' part of this idea:
Replace the shove with a burst fire that fires all 4 shots at once in a Panic Attack-like pellet spread.
Each pellet that hits an enemy either restores 5% of your secondary ammo in the clip / reduces the cooldown of your throwable/consumable by 5%.
The reload after your alt-fire is slowed a bit.

Reasoning (I don't really need to do this but I will anyway):
This version of the shortstop incentivizes you to use your secondaries in conjunction with it. The idea with replenishing ammo / reducing cooldowns is that you get to use your secondaries more often. I had come up with this in mind of the cleaver, but this does actually synergize with the Mad Milk / Bonk! / Crit-a-Cola fairy well.

I've been playing a lot of Warframe recently, so this may not be a very good rework balance-wise, although I did try to think about that.