I don't have experience with either of your options but I did go kind of whole-hog into the chassis gun thing. Bought a 338lm and an almost identical 308 as a trainer because it thought they'd be great. I haven't gotten a lot of time down on them yet, but I have to say, for hunting I really like a traditional stock much better. The rifles feel odd trying to shoot offhand, and it takes me longer to get set up on them comfortably. I hadn't done anything other than handle chassis rifles at gun shows before I bought them. I'm currently looking at good options for a traditional long range stock to try on the 308.
Regarding carrying it on a sling, get a stock or chassis that has the side mounted flush cups and you'll be good. I carried mine a handful of miles last season that way and if you put the sling opposite the bolt it's very comfortable
Thanks for your response and insight