How messy is filling with a syringe? I'm always worried about the ink spilling out of the converter, which may be a bit silly but it is what it is.
Unless you have Parkinson and even so... it is not messy at all. The only thing that comes to me is this: why syringe fill a converter? If you are going to do so, it is much better to syringe fill an empty double size cartridge and get almost double capacity.
If you try... what I'd advice as a cheap, quick and dirty solution is to get at your pharmacist an insulin/subcutaneous syringe (1ml) and an intramuscular needle ('cos it has a wider gauge). Fill up to, say, 0.8ml of ink, put the needle well inside the cartridge or converter and pour the ink in slowly to avoid overflowing and to check you are not making bubbles that might end up being ejected, popping and messing everything. That is easier if you place the needle touching the wall and let the ink slide in. When it is almost full (not completely) draw out the syringe. You do not want complete filling for then when you place the cartridge in the section, the nipple will displace some ink, which will overflow and mess around. Plus, it's not bad idea to leave some small amount of air inside.
When you are done, closed the pen and all, just flush the syringe/needle with water until it washes clean, this will ensure you don't get ink left to dry or to mix with other coloured ink should you want to fill with another ink colour later.
As for the OP's idea of fitting a needle to a converter... I find it kind of cool. Having always around a syringe or two for refilling, it is a convenient way to use those last drops of ink.
As for the advice on good/bad ink, I agree... up to some point. 'Cos, you see, at some point one has way too many ink bottles and wants to finish one, not to open a new bottle of the same ink, but to start a new, different ink, notwithstanding how excellent the ending ink was. In these special cases, one wants to use till the last drop and not leave it in the bottle until a new equal bottle is eventually opened somewhere in the unforeseeable future.