Most bicycles designed to fit in airline legal bags require some degree of disassembly to make it happen and removing the handlebars is almost always part of that process. If you're just looking for pictures of drop bar folding bikes, the Bike Friday Pocket Rocket and Airnimal Chameleon come with drops by default and I think people put them on changebikes. In general, the bars have a shallow reach/drop and sideways around the top tube and through the wheels underneath. I've also seen pictures of drop bar bikes with couplers. The Ritchey Breakaway is specifically designed for couplers but I've also seen them retrofitted on steel frame bikes.

If you're not especially concerned about folded size, I think the easiest way to get more hand positions is to put a loop handlebar or some other comfort mtb handlebar on the bike. There are lots of crazy shapes (butterflies, horns off the front, etc) but I use a shallow sweep loop on my Birdy. I matched the sweep with the flatbar sweep so the fold is unaffected but the front of the loop sticks out a bit on the side. The extra width is annoying when carrying it around but it doesn't impact my ability to take it on public transit. Relevant for your travel point, the bar is still a flat shape when off the stem. I find riding with my palms on the corners to be very comfortable, the front lets me get low, and I use the normal grip position for climbing and have no problem doing long rides on it.

I'd be a bit wary about putting moustache bars on a bike without a stem. Weight balance around the steering axis significantly impacts handling and in particular moving weight behind the steering axis tends to destabilize the bike. I don't know this would happen since bicycle steering dynamics are complicated but that'd be my guess as to what could go wrong and its' a bigger concern on folding bikes since they tend to have a lower trail.