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> I'm thinking something along the lines of a folding camping chair bag- something small and narrow that has a carrying strap.
You probably want something with a bit of padding.
FWIW, there's about a dozen companies importing various Chinese gun bags, and if Chinese manufacturers do anything exceptionally well, it's commercial sewing. All of the bags imported/labeled by the company Savior are also imported by other companies at often a lower cost, so poke around on Amazon and ebay, and if you're willing to wait, check out AliExpress and Alibaba.
I find it really useful to keep the entire kit contained - so barrel+BCG and the magazines all in one nice kit. If I were you, I'd lean toward a shoulder bag.
I only have the CMMG conversion, the BCG (and AR15 BCGs) fit damn near perfectly in this cheap LifeStraw case, and I keep that plus the 8 magazines in cheap shoulder bag.