H&R states in their owners manuals not to shoot milsurp ammo and no one has ever posted that it performs well in any of the NEF/H&R rifles that I've seen. Even the new 7.62x39 Handi does not shoot milsurp ammo due to the hard primers. The .223 Handis and Ultras come with a warning note stating not to shoot 5.56 NATO ammo in it, it is not the same cartridge as the .223 Rem.
The cheapest shooting rifle caliber that I'm aware of is the .223 rem, winchester whitebox 45gr JHP ammo is $13 for 40 rounds, $15 for 50 or UMC 55gr FMJ ammo at $5 a box of 20.....the WWB usually shoots best, but some have found the 55gr UMC to shoot ok.
I don't know how cheap 44mag ammo is, but I know 357mag ammo is real cheap, it would be the best pistol caliber that I know of if you can find a 357mag rifle, they don't make it anymore, but H&R has 357mag barrels available now if you want to send your frame in for it to be fitted.
Tim