The Bat HR is still in the 1800s for integral lug and rail, full nitride. The Vesper, which is an HR but machined out to be light is more and only available through unknown munitions. The ignitor is also an HR but with a bolt on rail, and integral lug. If you dont mind the bolt on rail, its an excellent value. I know its confusing. They are all HRs but they have different names. Bat is working on a new website that should help. They dont list all their options. You can get the HR with the rail cut out in the center like a GA hunter too. Bat makes a lot of different versions of their actions. Bullet central is the best way to order one. You will get it quicker through them than direct.
So far as the 338 choices, you cant go wrong if its a 33
I do have a personal preference for my 338 Lapua improved because it has just impressed the hell out of me. I have had 3 myself and chambered one for a buddy in Dillon. All 4 of those have shot groups in the 1"s at 1k yds. I have a bunch more out there and I have come to expect that from them. Another thing I like about them is for some reason they all shoot the same load. Its like a 6br. I can chamber one up and tell the customer to load this load, and he's 99% of the way there. Just something right about that case. My last one tuned at 2960fps with a 300 berger in a 26" Bartlien, 103.5 of N570. But the rum and edges are awesome too. In general the big 338s behave a lot better than magnums in other bore sizes.