I ran my TR4 for years at high altitude with a standard thermostat, no problem and no plugged bypass. It had good fitting aluminum radiator shroud and the radiator was flushed every couple years.
However, after hearing some tales of overheating, I had the radiator rebuilt and paid way too much for a sleeved thermostat. It happens to be a 160 degree. The engine is still being rebuilt, though, so I can't tell you how well it works.
I'd suggest just driving with the 180/sleeved and keep an eye on the temp gauge. If it's a problem, then start looking for a sleeved 160. Last resort is a "drilled out" stat, or a standard 160 with blocked bypass hose.
Be sure you have good fitting radiator shrouds.
Check that the proper radiator cap is installed and working. I think yours would be a 1", 4 psi version. (TR4s use 7 psi. The limiting factor is the heater core, which can withstand a little higher pressure in the 4.)
Another alternative is an upgraded water pump impellor, 6 blades I think, to move more coolant through the system. I think British Frame and Engine sells them.
TRF offers later (TR6?) 8 blade plastic fans, that I think would fit your car, but you already have an electric fan, right?
If you see overheating, consider an oil cooler, too (I'm currently adding one). But if you do, be sure to use one with a thermostat of it's own.
Consider an overflow tank, or maybe a true "header" tank, to improve fluid capacity and reduce airspace in the radiator.
Someone pointed out that 25% anti-freeze/75% water offers better cooling than the usual 50/50 mix.
"Water Wetter" can help keep temps under control somewhat.
Yeah, I've seen the heat in your area. I was in Phoenix during the record setting hot spell that closed the airport. 125 degrees, I seem to recall. We were installing a roof on a small building, of all things!
Re-reading your original post, urethane bushings will tighten up the front suspension. The only place they are needed is the upper/inner a-arms and it's a pretty easy installation (not necessary to fully disassemble... it's a good time to replace the ball joints, too, if needed.)
Urethane will ride a little rougher than rubber, and transmit more road noise. However, there are different "durometer" ratings of urethane, softer ones that provide a more stock feel, and still last a whole lot longer than the rubber ones which probably are shot within a year or two of installation.
The lower/inner control arms are nylatron which is harder than urethane and lasts a long time. The outer ones are brass bushings, also pretty durable.
So the only other place to consider urethane is the rear spring shackles, in place of the rubber bushings.
I've not done the R&P swap on a TR3. My TR4 has the early, solid mount style R&P and I always liked the handling (should like it even more now that the steering column and R&P are freshly rebuilt!) If you don't want to make extensive changes, I seem to recall reading about some possible improvements to the TR3 steering box, maybe in Roger Williams' "Improving TRs" book.
Cheers!
Alan