I have to say the .45-70 is the number one, always shoots well out of the box caliber in a Handi, but the .25-06, .270, .280, .30-30, .357 and .30-06 are all good choices too. The .243 and .308 seems to be a bit more troublesome, but even they are good shooters usually unless of course the superlight .243 is chosen. Usually the .223 standard and bull barrel are good too once the right ammo is found, but there's always one in the bunch that can give you fits, and apparently you're got one like that, there have been a few in just about every caliber, unfortunately, it's one of the most popular calibers in a Handi/Ultra.
If yours is a problem when shot warm, and that does sound like the problem from what you've said before, I'd do some work on the forend, or change to a wood stock, the syn stock for me hasn't been the most user friendly, it's harder to tune, even tho I like the weight, I have few syn stocked H&Rs, 3 centerfires and 2 rimfires to be exact, I like lams best, then wood, syn last.
Tim