I grew up on 2 wheels before the idea was mainstream. Its all I had to get around back than. There were no carts. I'd do it on a 10 speed but preferred mt bike handles. If you Minnesota bows saw a kid with a stickbow on the phalen range up to chisago, that likely was me lol.
I never did like slinging or strapping my bow down. Every time I did I'd have to bail and would watch that bow get ground into asphalt/dirt and what ever else was around. I preferred to carry it on top of my hands. Its crude but it works.
When I moved to AK I did the same thing on a mt bike and it worked well. I ended up with a bridgestone mt bike with some aero bars....yup I said that right a full aero bar set up. Its a complete oval, the front portion comes up. My bow would bounce all over the place and it was a pain, until I went to thicker wraps on the bars. Since I've come up with ideas to add to the bars (I still have that bike) but haven't had the need at this point anyways. A buddy dropped off a cart he found someone jerry rigged and broke. I think she's doable with some modifications.