I'm considering having a Ruger No. 1 rebarreled to .300 H&H. Would I be better to start with a 1-S or a 1-H? Do I want the additional weight of the 1-H profile barrel?

I'm shooting a 338 Win Mag in the S configuration and it isn't bad at all. Plenty of weight there to help suppress the recoil. One thing you might consider is the S version comes with a 26" barrel that will help wring out the extra velocity you may be looking for. Still more compact than a 24" barreled bolt gun.

Good point. 1-S it is then. Now to sell enough guns to actually buy the action...

I also have a Ruger 1S in .338 and it is one of my favorite rifles for balance and handling.
Ruger also made some 1S rifles in .300 H&H as a limited run. You might keep your eye out for one of those. Good choice on caliber! regards...

1-S is the way to go...1-H is much too heavy for the mild kicking 300 H&H, my favorite 30 caliber.

A 300 H&H with a cup full of H4831 and a 200 gr. Nosler is death and destruction on anything up to and including Eland...

One of my clients shot a couple of Cape Buffalo with his 300 H&H, he is very elderly and cannot handle a lot of recoil, but he can sure handle Buffalo with that old 300 H&H in his Rem 721 with 220 gr. Woodleigh solids and 220 gr. woodleigh softs..