Welcome!

Interesting way to carry PJ. Do you carry cotton balls too, or do you have another plan for having PJ turn into flame?

And how in the world did you clean the toothpaste tube out? Since you mentioned a syringe, I assume that you did it thru the normal hole in the end. If you did that, and are wrapping tape around the other end anyway, I wonder if you could cut the end of the tube, empty and clean it from that larger opening, then pack it full of PJ, using a table knife or small spatula. You would then tape/glue/somehow seal the open end.

I'm not too sure about putting the PJ on a burn, usually stuff like that traps the heat in, causing more pain...