Caltran[s] uses sign bridges with set heights. Their signs, for the most part, are the same height (I think there's two different heights) as the sign bridge or slightly taller. I don't know the dimensions, but I'm sure TheStranger or myosh will know.
That said, full-width tabs take too much space on the sign since they have a set height.
You are correct that the sign panel height should be the same for all signs on a structure (truss sign bridge, box-beam bridge, overpass, etc). In fact, when a truss or box-beam bridge is used, Caltrans goes so far as to specify what the sign height must be based on what they call the "frame depth". The frame depth is the height of the structure the signs are attached to. See the drawing below...
Of course, there are going to be exceptions (this is Caltrans we're talking about) but in general, the maximum height of an overhead sign is 120 inches. There are numerous examples of larger signs being placed on smaller structures.