You can do 5/8x24 if you can bear cutting down the barrel enough to get down to a bigger barrel diameter. I usually thread 9/16x24 for the Tikkas, you can go down to 1/2x28 for more shoulder. I've done several 6.5's at 1/2x28, it's not a problem. I've even done a few 7mms and 308's as 1/2x28 on...
As for the tight chamber. Are you using brass fired in another chamber and reloading? If that is the case you may need to reset your dies or run the brass through a small base sizer. The sound when firing could be related to barrel being so short. They are usually not as quite when using a...
I would avoid the the Browning. The triggers are not very adjustable and kind of a convoluted design not ideal for an actual trigger job. I would stick with 6.5 CM to streamline your ammo inventory.
I suggest the sig Kilo 3000, I find it much more efficient to rangefinder binoculars. Combining the two allows glassing with ranging. I would always have a reticle ranging back up cheat sheet. Sometimes with snow or rain, rangefinders just don't work.
22" barrel such as a Proof CF will provide good speeds with 7RM especially with a can because you'll get a little boost in velocity with a can. Tikka factory barrels are notoriously slow. However, a 6.5 PRC is much more efficient cartridge and factory ammo from Hornady is excellent.
I used it for 6.5 CM with 123 SMKs, got a 3,050 fps out of my 24" 8 twist Lija barrel. I used Starline brass and Fed 210 primers. It meters well and I had ES under 20.