Agree with the awkwardness of a mixed group of horses and bicycle. A horse isn't that fast when just walking, at least not for someone on a bike comfortably driving next to it. When my family had horses, I used to take my horse out into the fields on a lead while I drove my bicycle. When she walked, I also walked and pushed the bike along. I only sat on the bike when she trotted or cantered, otherwise it just didn't work.

But as mentioned, you could take a bike along if one character really isn't able to stay on a horse at all. If the horses are used to carry things, it wouldn't be too difficult to strap a foldable bike to a horse's back.