Not exactly what you're asking for, perhaps, but I've used a "backpack cat carrier" for this (basically a black backpack with mesh bits to see/ breathe through). I sometimes wear it on my front, which looks weird to other humans, but the cats tend to seem happier if they can see my face.
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I have a backpack for a cat as well. One benefit over bike-mounted solution is shock absorption. When i ride over bumps, i can stand on the pedals and the shaking doesn't get transferred to the cat as much as it would if it were mounted directly to the frame.
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.
Get a wide front rack installed (look for "pizza rack"), put the normal cat carrier on it with bungee cords and be very careful.
I stopped doing this and just started taking the bus because I felt it was probably too stressful for all, but that's just my opinion!