I would Google some vet recommendations.
I tried to find the video again but I couldn't. But I recently saw an Instagram video of someone asking a vet what they recommended as a safe method of transporting a pet on a bike. If I remember correctly - which I might not - the vet was strongly against something on the handlebars that the pet could be flung out of in a crash. I think they suggested a pet-carrier backpack style method instead.