Refilling carts has its advantages:
- cartridges can give better flow than convertors
- reduces waste through reuse of plastics
Ok, so maybe there are tens of discarded empty cartridges clogging our landfills but every little bit helps..

But I have heard it said, and my own obervations support, that cartridges, for whatever reason, are less prone to "vapor lock" where all the ink end up at the piston end of the convertor. Requiring a tap or even a twist to get things moving.
If a cartridge seal would break down over repeated uses, how much sturdier can a convertor seal be? I guess if you never removed the convertor when cleaning or changing ink colors that would help maintain the seal, but I'd guess an average cartridge could be refilled well into the double digits, especially if you're careful.
All that being said, my one piece of advice is use a syringe or thin pipette. You seem to have that bit in hand though. Using an eyedropper is not advised unless you enjoy water rockets and Jackson Pollock paintings...

Your fellow cheapskate

Tom