It's hard to say for sure if it's the tanks getting too cold, the reg is getting too cold, or the tank's valve is too restricting.
If the reg works fine with the big tank; then it's not the reg.
It would likely take the tank some time to "freeze up" (not that freezing has anything to do with it; drop below boiling point would be more accurate, but you know what I mean). One way to test if this is your problem is to time how long it runs on a outdoor temp tank. Then time how long it runs on an indoor temp tanks. 15-20 Kalvin difference should show you about 1/3 more time if the boiling point is really the problem.
If this shows no difference then the valve is too restrictive (which is likely) And two tanks will work for you. If freezing is the problem, than two tanks will likely work too, but two 20lb's would be better than two 10's.
Also, that gen is water cooled.... possibilities.... who needs a rad when you have a big AC unit...
OR:
get a really little propane tank and a very small burner, maybe a colman camp stove. Put this under a 20lb propane tank filled with wood chips. Then you'll need new regs that can take 350F gas. But after that just tweak the carb and your good to go!
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