I like the HS stock better than the BC stock but they are very similar for your rifle. Quality on the HS will be a little better for fit and finish but I don't know that you would see any accuracy differences. Whichever one you choose you should see an improvement over your present configuration.
As for the clamp on breaks, I am not a fan. I would have the barrel threaded 5/8x24 and a break installed. That would be your best option. I like the APA LB gen2 (I have a gen 3 as well but not sure the tuning ports are worth the extra cost). The APA LB is an awesome break, but it is extremely laud, deafening if used without hearing protection. I use them on a 300 WM and a 6.5 Creedmoor, both guns are set up identical to each other and honestly you cannot tell which gun you are behind, both can be shot all day with no issues.
A competent gun smith should be able to thread it for you for, as has already been mentioned, around $100 (my local guy, a friend, charges $40, my builder $150). They can also recommend a break for your intended purpose and some manufacture their own which they may include in the price to install at little extra charge. There are tons of break options out there for just about any budget, that work well. We have an OEM here in Alabama that manufactures there own break that is very effective and can be had for around $40. I have several on some ARs and a couple of 308s. I even at one time tried one on my 338, though they are not rated for that. They work fine. I'll post a pick or two of them, nice breaks for the money. My son has a PA break on his 308 I picked it up for about the same money on sale a few years ago. My 308 sports a Sure Fire, threaded for a suppressor which also works well ($90). These are just some option to consider, which ever you choose I am sure you will find it more effective than what you have and a great improvement to your old gun.
I think the Sig scope is a solid choice as well as sticking with your 300 wsm. I don't own a wsm or a weatherby any longer but for your intended purpose they will serve you well. Hope this helps!