No that has nothing to do with the cap.
The thermostat controls the minimum temp, not the max temp. The max temp is controlled by the capacity of the cooling system.
Over time aluminum radiators have something weird happen to them and the metal gets more brittle and becomes less heat conductive. The other weird thing that happens is the spot in the radiator directly in front of the fan clutch tends to collect more deposits and such so the thermostat in the fan clutch does not get air as hot as the rest of the air coming through so you can get a lazy fan clutch action and it not blow as it should when it should causing temps to rise at lower speeds. You can diagnose this by using one of those IR thermometers and checking the temp in the center of the radiator in front of the clutch to the areas around it. If you see a big difference that is what is going on. The only real fix is a new radiator.
Now when the system heats up enough to build enough pressure to vent the cap, you are talking way hotter than you are seeing and you will see it blowing out of the tank. Normally when you see this, you have a failed head gasket.
I really doubt you need a higher pressure cap. It is good practice every few years to install a new one, but at the same pressure rating as stock.
In summary, don't waste your money on a higher pressure cap.
G