bike trailer.. there are some for dogs .
i dont have any pet. I do have a Bike trailer.. carry freedom [city, in my case] uses a ball Shaped hitch
so the rear fold still holds up the bike as before ..

the CF,c includes a nylon bag, slings into the frame with 1 strap.

you could pay someone, with a sewing machine to make an alternate way to use the space..

UK based Carry freedom Y trailer uses a flat deck.. maybe just a bit of carpet on it would offer dog claw grip

+ you could fit a pet carrier box on top.