Why don't you just put the folding bicycle in a box and simply not tell them that it's a bicycle?
The particular folding bike I have has normal-sized wheels, so it tends to be obvious that it's a bike, even though it's not oversized luggage. I've never tried to talk it onto a UA flight, because I don't like the thought of having to choose between $350 RT in fees and finding a place to store my bike at SFO before a flight...
I'd consider buying a smaller folder if anyone reports reliable success with them. I really miss having a bike when travelling to the East Coast!