Poor kitty! He/she is lucky to have you.
I too have grappled with the cat on a bike question... One of our cats will allow me to do anything I want to him, and not really fight (or move). He's the type that would have placidly spent his life dressed like a doll attending fake tea parties if he'd landed in a family. Him I'd just put in the front basket and go.
The other cat we have will not tolerate any change or noise. All movement is evil and terrifying. Any restraint is a near death experience, and there is always a good reason to have a heart attack. No amount of coaxing or pre-exposing or trial runs would make any experience (like riding in a trailer on a bike) acceptable without heavy sedation. She once saw my bicycle in a corner of the basement that it hadn't been in before, hissed, and hid for four hours. I don't entertain the thought of biking with her anymore.
I guess I'm trying to remind you to be a good judge of your cat's personality and tolerance for new experiences. A trailer could protect the wee beastie from the elements, assuming you won't have figured something else out and this will still be an issue in the colder months.
Good luck, luckily it's manageable (both cats on bikes and diabetes
).