It would only be "overcooled" in a winter situation. In warmish weather there isn't much danger of that. IMO, "overcooling" is a remote possibility that most of us don't need to worry about. It's usually too much heat that causes worry. Still, to give the subject some consideration, I see it from two perspectives.
1) Viscosity. The trans is designed to operate most of the time with the oil at a specified viscosity ( just like the engine). When the oil is very cool, it will be significantly thicker and trans operation will be altered to some degree. That could make for harsher shifts and an increase in fluid friction that could decrease fuel economy to some small degree. There are mechanisms in the trans to moderate such effects but the trans was still designed for the oil to be at a specific viscosity (again, just like the engine).
2) Moisture. Moisture buildup is possible in an automatic and, just as with an engine, if it isn't baked out by the heat of operation, bad chemical reactions start to happen and damage can result. The very few times I've heard of problems in this area, and I had the 3rd degree the info out of a lot OE engineers and tranny repair specialists, it happened to people who short-hopped a lot in very cold climates. It's not a common problem but a possible one for some people.
My advice is to monitor your trans oil temps by installing a gauge and add whatever cooling solutions are needed. You want the fluid to warm up to normal temp (150-180F-ish) and stay that way no matter what. Some vehicles don't need any help to do that and some sort of gauge will help you know that. If you are in Las Vegas, you may be one of the few people who won't ever have to worry about overcooling, or at least only for very short periods.
The PCM in some cars will read trans temp and using monitors like the UltraGage or ScanGage (google them and check their application and features according to your cars), you can plug it into the OBDII port and monitor a whole bunch of engine and powertrain parameters.
You can install a regular analog gauge too. I assume your cars are those listed in your sig. I don't know those cars at all but assume they have acooler and cooler lines. FWD cars are often difficult to fit a gauge sender into the trans itself, but you can install them into the cooler lines.
Bear in mind that the temp reading you get from an automatic will vary according to the placement of the sender. It will read moderately cool in the pan... that's the "bulk oil" temp. The out line to the cooler will be the hottest, as it's bleed off from the converter. The return line from the cooler will obviously be the coolest. For experimental purposes, I had sensors in all three positions in my old diesel Blazer (700R4, a notoriously hot running tranny) and it was very interesting to monitor and see the differences. The pan could be at 180F, the out line at 240F and the return line 140F... all at the same time! The point is that you need to interpret whatever readings you get according to the sensor locale.
As to which is best? There are many schools of thought and I have vacillated myself. Since it's time-at-temperature that's the biggest cause of fluid oxidation, not those short high spurts of temp, I have gradually come to the conclusion that the bulk oil temp, in the main oil reservoir, the most useful parameter to monitor. You could make it work anywhere but if you only look at the out line, all you'll see are heart-stoppingly high temps from the converter. They will be highly variable too, only peaking when the converter is unlocked and you are working the trans. WIth the converter locked, the T/C is not generating much heat and you'll largely see bulk oil temps.The bulk temp is largely stable and increases and decreases very slowly.. making it a boring gauge to watch if you are into that (which I am!).
I wouldn't advise install any external cooler without a demonstrated need. You can only demonstrate that need with a temp gauge of some sort. Logic then dictates that a gauge is job one.