I very much doubt that it would be possible to have anything self-contained and small enough to work by electrical heating of the metal pin itself, within the size of the hatpin.
If you duplicated the pin using a stainless steel capillary tube (like a hypodermic needle material), that would allow something to the inserted or threaded through it.
There are then a few possibilities -
A wire through, connected to one of these just beyond the tube tip (with the tube as ground return)
Something like a miniature PTC thermal fuse - 1.6mm long x 0.8mm square.
With the right voltage, that will heat up and self stabilise at an appropriate temperature.
It would need experimentation to find out what temperature it reaches with what voltage.
A wire through plus a spark generator (from a cheap piezo lighter?) if that has enough energy to light the paper.
Or chemical means - a stiff wire "plunger" that can push a bit of a catalyst or reactive material - smoke paste? - to the end of the "pin" ??