I also have a heavy barrel H&R in .243. It also took me awhile to find a load that shot in it very well. I probably tried like ohhhh 30 different load combinations before I found a good one. I now have 4 different loads that work well in mine...but it doesn't mean they will in yours. This was one of the pickiest of all my rifles to load for...but it does shoot below 2" groups @ 100 now...the loads are...
80g Hornady FMJ, pushed with 38g of IMR 4064, Remington 9 1/2 primers...Winchester Brass
55g Nosler Ballistic Tip, pushed with 43g of IMR 4895, Rem 9 1/2 primers....Win Brass
90g Nosler Ballistic Tip, Pushed with 37g of IMR 4895, Rem 9 1/2 primer...R-P Brass
100g Nosler Partition, pushed with 41.5g of H4831, CCI primer....R-P brass
Not sure what loads you have worked with...but I definitely got the best results fro IMR 4895, H4831 and IMR 4064...I also had consistently better results with Remington Primers. I can consistently get under 2" groups @ 100 now (I know, nothing to brag about...but what do you expect for a rifle I bought new for under $180!)
All I can say is keep at it! If you want sub-MOA though...I recommend spending a bit more $$ than what that H&R set you back...don't get me wrong, I own 2 of them (223 and a 243) and like them both for what I use them for, but they are not going to be as accurate as my Remmy 70's or my Tikka...or my A-Bolt's...but I also paid 3-5X as much for them too.