It's good to see you here. I always enjoyed your posts over on shooters.com regarding LR hunting and see that you found it here like I did since the people here are also interested in LR hunting and won't slam you for aspiring for long shots. I used to go under the name bblaine2k there so you may recognize it.
Anyhow, to answer your questions. I am facing LR weaponry dilemmas much like yourself since I'm also relatively new to the sport. I'll share what I've been able to sift through.
First, in regards to how much energy and velocity is needed on animals, I've found the Optimum Game Weight Formula to be about the best approximation for determining whether or not a given caliber/bullet/velocity combination will work for a given game animal. For years, I used to go by the 1000 ft-lbs for deer rule but feel that this OGW formula is a better approximation and believe me that is all it is - an approximation. There's never any absolute certainties because of the wide range of variables involved many of which we probably don't even know about. So don't worry too much about it. The OGW is calculated as follows:
Velocity X Velocity X Velocity X Bullet Wt. X Bullet Wt. X 0.0000000000015 = OGW
A better way is to simply plug the numbers into the calculator at the following website:
Any of your mentioned choices 300 Win Mag, 300 Ultra Mag, 7mm STW, 7mm RUM should work fine on the animals you mentioned out to 600 yards.
As for accuracy between your choices, I can say this. I've handloaded 7mm STW to shoot 3/8 groups at 100 yards under less than perfect circumstances. The 7mm STW I was shooting was a rechambered Model 70 Win. that used to be a 7mm Rem Mag with a 22 inch barrel. Regardless, it still shot very well with CT 140 grn ballistic silvertips and 72 Grns of RL 19. The 300 Win Mag is another good choice and I've taken whitetails up to 525 yards with it - my longest shot in a Ruger 77 laid in a B & C composite stock. Never reloaded for it though but it still shot 1 inch or less at 100 yards. Also found great accuracy with 300 Weatherby Mags in a Ruger No. 1 that shot 5/8 inch groups at 100 yards - a rifle not reputed to be an ideal LR rig. The point is that most rigs can be tweaked with reloading and some good gunsmithing to increase accuracy as you already know.
For me, I've since bought a 7mm RUM in a Remington Sendero for my LR rig. It's about as close to a custom rig as I can afford. It has a long barrel - 26 inches, something many LR shooters have recommended to me. In fact some of the custom 7mm STW's sport up to 30 inch barrels! I chose it because even though the 7mm STW shot great for me, the 7mm RUM is supposed to be more accurate because it headspaces on the shoulder and not the belt like the 7mm STW, 300 Win and Weatherby Mags, and all other belted magnums.
Finding feedback on the RUM's has been difficult - most likely because it is a new caliber and not as many people shooting it like the 300 Win Mags and others. So even though it may, in reality, be much more accurate than the 300's, there's not been enough time for feedback. People have been shooting the 300 Win. Mag for decades so there's plenty of people to testify in favor of it.
Ballistically, the 7mm RUM is supposed to be one of the flattest, if not THE flattest, factory cartridge produced depending on who you ask! That was my main reason for going with it. Those against my choice warned me of the heavy recoil, and too much powder to marginally outperform the 7mm STW. Brass, as you already know, is only produced by Remington currently and it isn't cheap. 100 new cases run about $50.00. I haven't tweaked it any, haven't even shot it, since I'm still shopping for a scope, bi-pod, etc. If you're interested, I could let you know how it goes in a couple months to tell you my experience with it.
Don't rush into any gun purchase as I used to do. Keep researching like you have and make the best choice you can. I don't think you could go wrong with a 7mm RUM in a Sendero - I don't think I have!