Plumbing – Newly installed thermostatic mixer runs cold when shower head raised in height

If the cold water pressure is good, then you are right about the shower head not being the problem.

Check the hot water shut off valve feeding the tub to assure it is opened fully.

Also if you have good hot water pressure at the vanity and are sure shut off valve is fully open, the problem may be in the mixing valve that feeds the shower head. It's not uncommon for deposits to form over a number of years in this valve and restrict flow, especially on the hot side. Could also be a broken or stuck washer in the mixer. You can shut off the water to the valve, hot and cold, remove and inspect the cartridge from the front. Remove the handle(s), the bezel trim and the valve cartridge should come out with a wrench. If any of the internal parts look suspect, take it to your hardware store and they can help you select new soft parts to rebuild it. Replace it in opposite order that you disassembled it, and don't forget to put some plumbers putty under the top and sides (not very bottom) of the bezel, so water won't leak in behind it and down your wall.

Good luck.

Adjusting the whole-house water pressure is unlikely to solve this problem, simply because you set your shower temperature based on some initial balance of hot and cold. When other people draw cold water only, the pressure difference will still have the potential to affect your shower temperature.

Instead, you should get a pressure balanced valve or shower fixture. These fixtures automatically reduce hot water pressure when cold pressure falls, so instead of a temperature spike you simply get a momentary decrease in output pressure. You can find pressure-balanced fixtures (which would replace your existing shower fixtures) or an independent pressure balancing valve (which you'd want to install near the shower but would allow you to keep your existing fixture).