I tend to think of the .300 H&H as a warm .30-06 tho I realize it can be loaded to almost 300 Win. Mag levels and I have seen such loads.
I once had a .300 Win. Mag and wow that thing shook my teeth loose when I fired it. The gun was a Remington M700 with an almost pencil thin barrel and light synthetic stock. It was way too light for me to be shooting as a magnum. I fired less than two boxes of ammo and traded it on same rifle but in .270 Winchester and sure never missed the magnum.
However, in answer to your question, yes I think the 165 Accubond should be an excellent choice for you in your rifles. I tend to go with the Nosler Partitions mostly myself but a 165 or 180 of either bullet should do all you need them to do.