I can offer four pieces of advise:
1.) One experience of taking a bull moose w/ a 375HH; not an expert
2.) Which handloaded bullets have shot the best in my rifle
3.) If I had to start over today reloading the 375HH
4.) Factory ammo guidance, based on the above three items
I took my bull moose w/ a handloaded 260 accubond leaving the muzzle at 2,715fps from a 24". Moose was 200 yards away. First shot hit him in the shoulder; shattered one front leg, but he was still wobbling on three. Second shot was perfectly in the boiler room, and he face planted / lights out. Impact velocity was ~2,300fps at that distance. First bullet fully disintegrated when it hit the shoulder bone; second bullet was on the off side and did not fully punch through. Did the bullet work? Yes. Do I still shoot that bullet? No, I would have preferred additional penetration.
Despite gun magazine article talking about how the 375 puts all bullets in the same hole, that is not my experience with my rifle. I have found the 260 partition shoots slightly better than the 260 accubond. I have also found that the barnes 250TTSX and 270TSX shoot well with various medium powders (IMR-4064, RL-15, Varget). The hornady 300 DGX shot horribly for me w/ several pounders.
If I started over, I would shoot barnes 250 TTSX bullets 99% of the time. The other 1% would be barnes 270 TSX for things trying to eat me. While RL-15 will give you good accuracy and velocity for the 250-270 pills, varget behind a 250 barness TTSX is fantastic. ~71 grains slings them at 2,825fps and prints nice 3/4" groups. The case fill is absolutely perfect - no powder rattle. Velocity, accuracy, and point of impact do not seem to change w/ temperature.
Double tap ammo offers the barnes 250 TTSX. I would call them and ask when they expect it to be back in stock.
link to double tap ammo My 2nd factory ammo option would be 260 partitions from nosler. Currently out of stock and expensive as hell, but they shoot well and its a tough bullet. Third I would try the hornady outfitter 250 CX. I have not tried this bullet, but would hope it would perform similar to the 250 barnes. And if I coudn't find those, I would just buy Hornady 270 spire point (in stock). Factory ammo in 375hh has gotten extremely expensive; if you are a handloader, you will have more options and shoot more. Midway has factory overrun 375hh brass for sale at a really good price.
brass link And barnes 250 TTSX are in stock at multiple places. If you don't have Varget, others such as RL-15, IMR-4064, or even IMR-4895 will give you good results. For the cost of 20 rounds of factory ammo, you can buy 50 pieces of brass and bullets.
Hope that helps. Message me if you would like a summary of my handloading notes.