Hi Ave.,
First off, 12 ga or 20 ga? The 12 ga USH frame will only accept the 12 ga USH barrel or a 10 ga shotgun barrel. The chamber swell is wider on the 12 ga USH barrel so the frame rails or relieved to accept it.
The 20 ga USH is on a standard SB1 frame so it is only good with shotgun, muzzleloader and pistol caliber cartridges like 357M and 44M.
A NIB 308W will have an SB2 frame and you should be able to send that frame into H&R and have a 20 ga USH barrel fitted, assuming they have them in stock. You would need to verify the age of the receiver on even a NIB Handi because sometimes they are forgetten in a warehouse for many years and may be many years old. The problem is SB2 receivers made before 2000 or 2001 (check the FAQs sticky thread, I have CRS
) have limitations on what barrels H&R will fit to them.
FWIW, if a shotgun is legal in your rifle zone a properly setup 12 or 20 USH with the right ammo can be a legit 200 yard deer gun so you might be able to get by with just the USH depending on your hunting area.
Good luck and thanks for reminding me that whitetail archery season opens for me in less than 2 months. Time to drag out the Xbow.
BB