I have been fighting 300gr XTP's in a 77/44 for 20 years. I have an integral 77/44 from the old John's Guns, John Tibbits from Palestine TX. I have 2 spots that have a 35 yd and a 53 yd shot for hogs so I am not worried about 100+ yard accuracy.
John's recommended load was 300gr XTP, 14.0 gr W296, WW case, Fed primer. I will think I have it shooting 3/4" at 50yards and then it or me pulls a wild one that opens up to 3-4". I have even tried shooting no more than 2 shots in case I am getting some thermal expansion in the suppressor. Still no joy. I will shoot it for a while and put it up for a year or 2, get the itch again and try again. I have tried 10-15 powders with no luck for a repeatable group. I have tried large rifle std primers, all case prep and trim on the brass, etc. The only thing I haven't done yet is free float the suppressor. PITA.
The factory barrel on a 77/44 is 1:20 (new listing), the older guns have been listed as 1:16.
Everything I read says try 240-250 gr with a factory barrel. Lots of people say they get good accuracy with Trail Boss. I have never gotten that powder to shoot. I had no luck with the lighter bullets either. I won't shoot lead in this because it has a ported barrel and can't be taken apart.
I picked up a 77/357, SBR'ed it and it will consistently shoot 190gr lead at about 5/8" @50 and it is a blast to shoot!