I don't think you will find a "quality" pack with the straps you wants. Honest opinion.
You could try a Eberlestock with their thinner straps, but that's about all I can think of. Past that, my advice would be get used to shooting a little bit of a thicker strap. It's not like any quality pack being talked about has super thick straps. I can take a quick shot with my Kifaru if need be. But again, not quiet in the brush.
Other than that issue, I would say Kifaru frame with only a Guide Lid and a Grab It. No pack. The Guide Lid will hold more than your fanny pack. Then you strap the meat in game bags onto the frame with straps and the Grab It and lid holding it all in. Shoot you could even roll the Grab It up and put it in the lid until needed when you shot something. Then the only thing you are trying to keep out of brush is the small footprint of the Guide Lid. Or run just the Grab It, normal lid and straps for meat and run a Hill People Gear Kit bag for your gear. It would probably be comparable to your fanny pack for space.
As far as you not hearing of HPG, they are a known company for their Kit Bag especially. That's what those two brothers are known for. Call them up. I've had them have to pull over to answer my call but they do it and placed my order when they got back. I really like my kit bag. It's just a small company rolling out quality products.
https://eberlestock.com/products/x2-pack
https://store.kifaru.net/guide-lid-p172.aspx 1200 ci. Check out the picture of the dude wearing his as a backpack with the straps that are included with it for an idea of size.
https://store.kifaru.net/grab-it-ii-p24.aspx
http://hillpeoplegear.com/Products/CategoryID/1 When I say as much room as a fanny pack, there are three compartments to a Kit Bag so that would be with each one expanded out. These pictures don't have them expanded out.
