There are fountain pen shops online that sell such syringes, and I've read claims on the Fountain Pen Network etc. that many simply use ordinary medical syringes to fill proprietary cartridges with their preferred ink. Before I go this route I'm going to try other brands that either have or are compatible with refillable bladders; the Kuretake No. 8 Brush pen costs about the same as the Pentel, and can be used with a Platinum converter and Platinum carbon ink. I'm also not terribly committed to brush and fountain pens - I've never found my ordinary brushes and dip pens that inconvenient to carry around, and stopping to clean the pens/brushes gives my wrists enough of a break to keep carpal tunnel at bay.

Thanks so far as the drawing; it's somewhat based off my girlfriend's rottweiler