You could run the 375H&H case at 40,000 in the Handi but no more. NEF chambers the 450 Marlin which is essentially the 2" American, but the ammunition is only making 40,000 PSI. Pushing it harder will no doubt streach the frame as has been reported by one fellow who did a rechamber to 300 Win Mag.
40,000 PSI loads in the 375 case would be nothing to sneer at, 300 grain bullets at 2200 fps or so, basically starting loads for the 375 but that is 45-70 performance. Problem is that the same performance could be had rechambering a 38-55 barrel to 375-06, but then you would have to address the .379 or so barrel people report the 38-55 having rather than a 375 barrel, unless you were talking a rebore.
The best way out would be to invest in an Encore. It is chambered in the 375 H&H and will handle anything that is based on that case. As well, it would be no big deal to get SSK to make a barrel for it, and I think you would end up with a better gun for the task in the end.