It's a little more complicated, but may be the easiest solution to your problem.

Buy an UNvented bladder...I think 12 gallons is about the smallest these days. Or any small cheap rigid tank...but you'd have to vent anything rigid. However, that's not much of a problem 'cuz it's only water. Find a convenient location between the toilet and the sink into which you can stuff it Tee the fill into the sink drain using a y-valve (Since this is an above WL installation, you could even use a plastic $3.95 garden hose diverter here) and the head intake into the discharge. That's all the plumbing needed. To use the sink normally, keep the valve open to the thru-hull...to fill the flush water tank, turn the valve and run water down the sink.

This also works on sinks that drain below waterline...for those who want to flush with fresh water all the time.