Last summer in the Phoenix area, my shower was at a trickle before I could shut down the hot water enough to have a shower without getting scalded. I remembered when I lived in Yuma, the showers had Mixet valves. I didn't think anything of it until I didn't have them.

I want to remove the single valve and replace it with a Mixet.

Is this something a DIYer can do?

it depends on how big of a hole you can make in the wall.

Any brand of temperature balancing valve will work. In my area I recommend Moen because it's so easy to get free replacement parts and they are so easy to fix.

"DIY" leaves a very broad spectrum of skill to assume. If you know how to sweat a joint and have the tools to make a hole that's big enough to remove the old valve and put in the new one, then yes. But depending on the type of piping used, sometimes that too is difficult even for a pro. If you have access to the opposite side of the wall, that's usually what we do. Means the only repair needed is drywall and you're guaranteed to be able to get the new valve to fit in.