In this construction, I want the vertical hinge to be in this location, centered in a 3 brick wide area. On top of it is two plates and layer of studs that match the 5:2 ratio, so in theory the hinge would be secure in this 3 brick wide gap. Is there a more elegant way of securing both sides of the hinge? I've looked at some bracket pieces but the issue is that they all have studs on top rather than stud-holes.
I would remove last two plates since they ruin the geometry here. And instead I can suggest using any modified brick with two studs facing towards hinged brick (white colored bricks). Technic, Pin Connector Round 2/3 L (seen in red) could then be used to attach studs on both sides. Orange plate below confirms all of this fit in required space.
A bit of background on the logic with my solution. With cases like this it is helpful when you measure everything in standard units. Here, brick height is 6 units, plate height is 2 units and stud is 5 units. From here you can clearly see you have a gap of 3 studs=15 units and your assembly is 14 units. The difference in 1 unit makes things more complicated. So my suggestion is to simplify it and make use of more common pieces.