What weapon upgrades make their corresponding proving ground significantly easier?

I can't remember for sure, but Power String should help with the Breaker's Bow proving ground. Impact Shells and Expanded Chamber can make the Scrap Musket challenge trivial.

Beyond that, I'm at a loss for now.

Generally, for the challenges it's best to spec for the speed/reload upgrades as opposed to damage (The Calamity Cannon and Army Carbine in particular benefit from this).

The Werewhiskey spirit, which gives you 100% critical hits, helps a lot on the repeater, pistol, bow, and pike challenges ... and possibly others. Before starting the challenge, jump off the ledge to lose health until it activates. Then you'll be able to one-shot everything.