My dad had one built on a SA Rem 700 about twenty five years ago. To the best of my knowledge, all he's ever shot in it are 75gr Bergers and I'd have to double check but I'm pretty sure he drives them with a healthy dose of RL25. I've actually shot it more than he has and it shoots VERY well. I've killed countless coyotes with it over the years and before I had some of my own varmint guns built it lived with me.
A short action will work fine but it needs to be throated accordingly or you could be limited to shooting it as a single shot. Not like it's lacking for case capacity so seating your bullets a little deeper for SA use isn't a deal breaker but if you're looking to squeeze ever ounce of velocity out of it a long action will be a better choice.
His has a 26in. Lilja 1-9tw barrel and even though it shoots the Berger 75's very well I think it's borderline and your 1-8tw is a better choice.
As you have, likely, discovered finding Rem 6mm brass isn't the easiest thing to do and unfortunately all of it is just "ok" quality brass. You can always find it on Gun Broker but it's usually at a premium price. I'm sure there are MANY TTH/22-6mm shooters that would be thrilled to see Lapua make Rem 6mm brass. lol
I don't think I'd even consider building a 22-6mm until I was able to acquire a few hundred rounds of new brass though.
As good as my dad's 22-6mm has been, over the past twenty five years, when the day comes to rebarrel it, it probably won't get chambered back to being a 22-6mm, it'll more than likely come back as one of the many variants of 22-243 or maybe 22x47 Lapua but if a 22-6mm is what you want, you certainly won't be disappointed in it and with your 1-8tw barrel you could easily shoot the 80's if you wanted to. But, I'd get acquire the brass so I knew I had it before I set off on having one built. Best of luck to you on your build.