Rd,
The yerushalmi kappelach are not at all like the NaNach headgear. Not on shape size or way they are worn. The NaNachs are more similar to the Bucharian yarmulkas in size and shape.
And it’s not a “few” chasidim. The Yerushalmim wear them. They put on tefillin just like anyone who wears a normal sized yarmulka does. A better question is how do those who wear tiny little kippot put on teffilin, they have almost no head covering at all.