I work in SalesOps in a Biotech (data/reporting). At one point, I did work from the road, only it was part of a climbing trip in a van, not a bike trip - but somewhat similar. I found working actually *in* the van didn't work for me - it was too cold, I needed reliable electricity and internet and didn't really have it. So I would work a few days in a row at the library (in Moab) or a hotel, and then climb a few days.
You need to evaluate what you need to do while you are actually working - do you need internet? Is public wi-fi going to work or will your company's security block that? Do you need phone? Is your laptop screen & mouse sufficient? Do you need to work during normal working hours, which means you will be riding only half a a day or not at all when you are working? Do you know in advance which days you will need to work (feasible), or will you need to be available during large portions of workday? (not feasible)
My first tour, I tried to get a leave and they didn't grant it, so I quit the job and when I returned I got temp and contract jobs... I've now been a contractor for 10 years and am about to go back to being a full time employee. Being a contractor is great if you have skills and a network. I had a career angel who took me along with him to several contracts, that was great. If you aren't marketable, don't have a reliable network, or the idea of looking for contract work many times doesn't appeal to you, then you might not want to embark on this path. I didn't ever think I would be a contractor for so long, but it worked out and I loved it.
Be careful asking for a long leave, asking can open an exit door if things are otherwise shaky. I don't think 3 weeks is a long leave, though, it's just an extended vacation. Before I quit that job, I had several of those - I just asked for a week unpaid leave to add to my vacation time.
Good luck!