There are a fair number of bullets in 0.311 and 0.312 to choose from, including the 150gr TSX. Handloading for a handi you could probably load a bit hotter than you could for a No1, No4 or No5 Lee Enfield ( P14's and Ross rifles are stronger, but less common so factory loads are a bit light, similar to the way factory 45-70 loads are). Then there is also the 303 Epps Improved (similar to Ackley Improved, but developed by Ellwood Epps), which provides ballistics similar to 308 Win. I think a 303 Brit Handi would be pretty cool, but I am Canadian.
Jim
BTW Epps is still in business, and still does chambers for the Epps Improved. They might be able to set you up with a barrel for a stub project like PhatInjun mentioned