The airplane bit is a big blocker for me. I've yet to do a successful dropper converter that didn't explode on a plane for me.
Forum responses are usually vague and along the lines of "try different viscosity ink," which, yeah, but there's tens of brands and thousands of flavors.
-I've flown with fountain pens several times without incident (though I will admit to being slightly nervous!)
I put the success (thus far) down to always filling the pen all the way before flying, and also making sure the nib is pointing upwards at all times.
That being said, there's no denying just about any pen is better suited than a fountain pen for flying.
Edit: pens I've flown with - Lamy Logo and Dialog 3, various Pelikan Souveräns, the occasional Metropolitan.
Edit: Presumably obvious, but still - the reason for filling the pen all up prior to flying is that ink does not expand as air pressure decreases - unlike, well, air. If there is no air in the ink reservoir, outside pressure is more or less irrelevant.
I don't think I've flown with a homemade dropper-converted pen. However, I've flown with (and used on the plane) a fully-filled Moonman M2 (~$25 on Amazon), which is built as an eyedropper pen, and I enjoy writing with it.
Forum responses are usually vague and along the lines of "try different viscosity ink," which, yeah, but there's tens of brands and thousands of flavors.