I think there might be some confusion about what the intended installation will look like. My understanding is that there is an existing drain tube from the sink through the hull below. The intention is to add an intersection to that drain tube and pump from the bilge into that tube.
If that is the case, there is absolutely no benefit to the vent. The only way it could possibly have any benefit is if you closed the sink drain with an airtight plug. Otherwise, the sink drain IS the vent.
The second part of this is the issue of how high the bilge hose needs to run. This is simple. In a worst case heeling attitude, draw an imaginary line the shortest path from the through-hull to the bilge pump, following the hose(s). If this line does not go above the waterline, there is a risk of taking on water in that attitude.
But, there is a third part to this that DJ addressed. If that intersection is below the waterline, you can still establish a siphon, even though the sink is a vent. That is because if there is no point along that imaginary line you drew which comes above the waterline, where air can enter the system, you can establish a siphon.
So, if your intersection is at a point that is safely above the waterline, then the sink does act as a vent, as there is a point along that imaginary line which comes above the waterline, where air can enter the system. If not, if you do not add a vent, you are at risk.
If that is the case, there is absolutely no benefit to the vent. The only way it could possibly have any benefit is if you closed the sink drain with an airtight plug. Otherwise, the sink drain IS the vent.
The second part of this is the issue of how high the bilge hose needs to run. This is simple. In a worst case heeling attitude, draw an imaginary line the shortest path from the through-hull to the bilge pump, following the hose(s). If this line does not go above the waterline, there is a risk of taking on water in that attitude.
But, there is a third part to this that DJ addressed. If that intersection is below the waterline, you can still establish a siphon, even though the sink is a vent. That is because if there is no point along that imaginary line you drew which comes above the waterline, where air can enter the system, you can establish a siphon.
So, if your intersection is at a point that is safely above the waterline, then the sink does act as a vent, as there is a point along that imaginary line which comes above the waterline, where air can enter the system. If not, if you do not add a vent, you are at risk.