All the other answers seem to be trying to solve the bike problem (buy him carbon offsets) or tell you how to convince him to change his mind on the bike issue. I suggest you utterly ignore the bike.
Ask yourself: if this employee was parent to a small baby, and objected to going to the other office because the baby would need to have longer days in daycare, and objected to the travel because "I don't want to be away from my baby for that long", how would you react? What if this employee was caring for an aging parent or a spouse that was dying of cancer? Substitute the entirely voluntary bike thing for something that is less voluntary and that more people feel very strongly about. Something you can perhaps relate to yourself.
Some employers would say
I'm sorry, but this kind of travel is a condition of employment here. You can find a way to make the travel work, or we can change your job to one that requires no travel but [pays less, is only 30 hours a week, doesn't involve all the same duties], or you can take a few weeks to find a job somewhere else and we'll wish you good luck.
(There may not be the middle thing about changing their job to something lesser, depending on the company size and jobs that exist there.)
Others would say
In that case I will only send [your coworker] on these trips. Please don't be surprised or offended when that coworker is promoted before you or gets a larger raise from you; you're deliberately choosing not to contribute to something important to us.
What would you say? If the reason was personal, strongly held, supported by most of society, and not considered "weird" or extreme? Would you still say, in effect, "don't let the door hit you on the way out" or would you find a way to support that decision not to commute or travel more than a small distance?
If you would say it's a condition of employment, there's no-one else to send, and if I need to hire someone else who will do it, I can only do so by parting with this employee, then say that. Don't let the bike thing confuse you.
If you would find a way for a "good" reason then stop judging reasons and find a way. Not asking other people to do more work for less money, or do all the travel while being rewarded the same as the nontraveller, but a real true fair way to support this very firmly held desire. I don't think you need to involve the biker in that process much, unless someone says "I will do all the travel if the biker does all the X" and you need to see if the biker is cool with that. This could be a very powerful retention strategy for a valued employee, since the chance of finding another employer who could accommodate this belief system may be slim. And all the other staff will see that you are flexible and generous, which is generally a good thing to be seen to be.
Note: I am not suggesting that caring for a family member, whether newborn, aged, or ill, is comparable to the bike preference. If you would fire the employee even if this dispute involved a clearly "good" reason then go ahead and fire them over this. It's not about the bike, it's that travel is a condition of employment. That half of the advice is easy. The second half is harder. If you would find a way to make it work, a real true fair way, not just asking others to do the person's work for nothing, then there is a way to make it work. So even for this reason, why not do it? Why not be fair?
As for the biker giving you the cold shoulder, very shortly either the biker will be gone or you will be delivering the news that "you win, you don't have to travel far any more" and either way your cold shoulder problem should stop.