I understand what you are trying to simulate.
Well, the sea level elevation due to tidal forces causes tidal currents in these regions, because the water will flow in/out with a velocity that depends on the velocity that the surface is moving and the section area of the inlet (that is, with a lot of simplifications).
As far as I can see, from the technical point of view, for your simulation you only need, as boundary conditions, the surface height and the velocity components at the inlet.
About your question "The question I pose is this: if the boundary needs a velocity field...". I think that, as I said previously (and Jean-Michel said
here), that only the surface height is not enough for the system to be solved.
If your simulation is experimental (not based on real events), then use the mass conservation principle to calculate the inlet velocity field. Otherwise, try to get this data from somewhere (I use HyCOM).