Hi Wishful thinker,
With the H&R you need the SB2 frame and the 10, ga has a bigger diameter barrel for bigger cartridges like the .577 cartridges. The new barrel is made by a gunsmith using the "stub barrel" method which employs 6 inches of the old barrel.
My H&R 577NE weighs 12 lbs. using a 26.5" bull barrel. I made a stock with nearly no drop at the comb/stock, best for recoil. ( Not bad to shoot with moderate/average loads.)
My TC Encore is barrelled with 500 -110 winchester (50-110 Winchester necked down to there .500 calibre.) by MGM Barrels. I think that they would do a 500NE also.
Make it weigh 10 lbs or use a muzzle brake. It's easy to router out the forend and fill the space with a mix of lead/epoxy. This adds about a half pound.
The traditional pressures of these old cartridges, as you know, is low so it is no problem for the weak break open rifles if you load wisely. I think that the mid 30,000's is correct, but do your own research, don't follow my guess work on loads. Brian