I would think a cat/small dog travel duffel bag lashed to the top of your pack would work out.

something like this: https://www.chewy.com/outward-hound-travel-airline-approved/dp/217290

You would want to make sure it is securely attached. The arrangement would make the pack pretty tippy. I wouldn't put anything in the top lid so you can keep the dog as low as possible. I also learned from putting my kids in hiking backpacks to always pick-up by the top middle loop on the pack (not shoulder straps).