As long as it's the head sink, not the galley sink and the toilet INTAKE that shares the thru-hull with the sink drain line. Several mfrs plumb that way to eliminate a thru-hull...Tartan used to.

Tee the head intake line into the sink drain line as close to the thru-hull as possible...below the waterline. That will make sure that the head intake can always prime. To rinse the sea water out of the system, just close the seacock and fill the sink with clean fresh water...flush. Do NOT, however, use gray water from the sink as flush water...wreaks havoc in the toilet pump and causes problems in the holding tank.

Btw...USCG regs require that all waste tanks--black and gray water--be vented...not only vented, but vented to the outside of the boat because methane is flammable and anaerobic sewer gasses--sulfur dioxide and hydrogen sulfide--generated as waste breaks down without ANY oxygen are toxic, even lethal. If an unvented bladder has been overlooked in a USCG inspection, it's because the inspectors were either poorly trained or just missed it. I strongly recommend you correct it because gasses generated in the tank pressurize the tank and bladder tanks are highly prone to blowing out fittings when pressurized. That only makes a really nasty mess but is also a serious safety issue.