The .308 choice is great, and the ASA choice is great.... but they don't come together. The ASA is for small calibers only (5.56, 5.45, 6.8, 7.62x39, 300Blk...)
The only side charging .308AR's (off the of my head) are the LWRCi REPR and the FA Titan... OR any custom work you get done... I'm pretty sure both of those options are reciprocating.
If you want a .30 Cal ASA build, check out the 7.62x40 Wilson, the 7.62x35 (300Blk/300Whisper) and the infamous 7.62x39 Russian.