I looked into folding bikes, and determined that for my use, a regular road bike works fine.
I use a regular road bike with quick release axles. If you use a mountain bike style with straight handle bars, they fit into smaller areas than 10 speed type bars. Single speeds or internal gears like the old 3speeds everyone is familiar with (available in 3/7/8/14/or the nuvinci infinitely variable speed versions) make for quick and easy wheel removal.
With the bicycle market today, you can buy an aluminum or carbon fiber bike for $200 at walmart, or $400-$700 at a bike shop, or a good used bike off craig's list for under $200 and rebuild to suit for well under that. mountain bike style is typically smaller all around than a road bike and already comes with the flat bars, but a road bike will be lighter.
whichever way you go, get high pressure tires, and smooth tread, mountain bike tires with aggressive tread patterns are a real pain to ride any distance on hard surface roads. If you want a road/trail tire, get the combo tires, smooth center with knobs on the edges.
My road bikes with 10 speed style drop bars easily fit in the cockpit lockers just by removing the front wheel. But being an outboard in the well version, I have tons of room below that would otherwise be filled with engine, so locker filler can be placed below.