I can't address the regulatory issues of STCs and the like, but I can address the performance issues of loading a 172 variant (whether on the 172 TC or the 175 TC) to gross weight, because I've done it with a lot of different years, versions, and modified aircraft. The highest powered 172 variant I've flown had the STC'd 210 hp Continental, which in reality just allows the governor to be adjusted to permit a higher rpm. The lowest is, of course, the 145 hp Continental. In between, I've flown several 180 hp versions, including a Cutlass RG, in which I probably have 45-50 hours. The most common power increase is the 180 hp Lycoming, both with a fixed pitch and with a constant speed prop. My own is a P172D, on the 175 TC, with a 180 hp Lycoming and CS prop, which I've been flying for nearly 15 years and about 850+ hours.
Here's what I know, from flying all of these 172 variants. They are all pretty much dogs at or close to gross weight--and that's at their legal gross weights. Sure, the higher powered versions climb better and get off the ground a little quicker, and they're able to climb to a higher altitude. But those are only incremental improvements, and their flying characteristics are much less pleasant, than if they're flown at as little as 75-100 lbs under gross. To me, most 172s are really pleasant to fly, under gross and with the CG mostly centered. Load them to gross and at an aft CG (within the envelope, though), and they become twitchy, almost too light on the controls, and much less responsive to control inputs.
I'm not going to admit to flying any 172 (or any other airplane) over gross, although it might have happened at sometime in the distant past (and certainly long enough ago that any statute of limitations has run

), but I've flown my own and many others near gross, most out of high elevation airports. Adding an additional 200 lbs, for example, would cause performance to suffer so much that the airplane would have really marginal capabilities. IMHO, it would go from being pretty close to the safest airplane flying to a marginally safe airplane, perhaps even unsafe under some conditions.
To me, the very best way to obtain a gross weight capability of 2750 lbs. is to move to a 182. A 182 flown at that weight is a really good performer, stable, solid, and capable.
Cary