I was thinking of upgrading some of the hardware on my guitar for Christmas. I have a Michael Kelly Patriot Custom. I was thinking a new bridge and new tuners, mainly because the guitar gets way out of tune on bend-intensive songs, such as Time (Pink Floyd). My G and B strings are way out of tune by the time I finish playing the song.

I'm pretty happy with the pickups in the guitar now, Rockfield SWC Pickups. They are coil-tapped, which gives me a wide variety of tones. I'd only consider new pickups if they were also coil-tapped

What are some upgrades I should consider for my guitar? I'd say the budget is $200 max.
Aside from coil splitting, you could add in a phase in/out switch. The cost is quite small in terms of money, and the only other thing would be drilling a hole in the guitar. Replace the bridge as well (Not the tailpiece, just the part with the saddles), maybe to a Gotoh or TonePros. For the tuners, look into Schaller, Sperzel, Grover, and Gotoh, and see which ones you like most.
It would be cool to have a guitar shaped as Christmas tree with Christmas colors. hehe. The head of the guitar is the stump, the neck is the log, and the body would be the pines.
What would a phase in/out switch do?

You recommended Grover tuners and a Tonepros bridge, but the guitar already has those. Are there better parts from those brands, or am I SOL?