We have decided that the cheese in the middle is really superfluous, but adding a bit of cheese to the meat mix seems to be a good thing. We'll try that again some time.
Meanwhile, the technique that I've put together for deep frying food in our big skillet seems to work really well. You fill the pan without about half an inch of oil, bring it to heat, and then add the thing that you want to fry, turning occasionally so that both sides get some time dunked in the oil. We did this with tater tots again tonight and they were lovely, crisp, and not very oily in our assessment. Nor did they get soggy as they cooled off.
This system takes about twice as long as normal deep frying would (which I suppose is not a surprise), but you don't have to deal with maintaining a deep fryer. The oil has been decanted back into a jar and is back in the fridge for storage for the next time that we want to try this trick.
Even K liked the tater tots. I can tell, because she finished the last of them. :)