For good or bad I don't do chain cleaning ever. If I go onto travel I cannot take any chain cleaning setup with myself anyway and will not busy myself with cleaning when all kinds of other stuff piles on me in travel. By now I made some straps myself but all that mess had no thought put in as far as handling a folded bike was concerned. At the time I bought the bike, the folded mast option was not available and the design concentrated on dismantling the bike for suitcase - that was semi-OK - but not really tackling the folded bike such as for car trunk. I can design/tweak things myself but you want a starting point where you are half way there and BF is just where you throw your hands up.
Riding BF is generally OK. Low points is that the steering is squirrely and in steep uphill or downhill - don't remember already which - I was afraid I could flip over. The steerer tube-stem interface scares me whenever my mind comes back to it - something I cannot quite fully forget about while riding.
FYI since I started using squirt lube I have never cleaned my chain. You just reapply as needed, no muss no fuss. It dries and doesn't get on anything.
You must have a very old BF if the folded mast wasn't available. Thousands of folks ride them and don't have the issues you are thinking of when riding, but whatever makes you comfortable (or not). Happy rides =)