I was hoping someone might have kno

You may find what you're looking for in King's Handbook of Hydraulics. If you live in the US you may find it at a local library.

Sounds like a Cipoleti weir, but that is sharp crested. Check out an equation developed at the University of Missouri, Rolla, - Wilson, Scott, Wolf. The following equation was developed to model the discharge from a weir like you are describing.

Q=0.86H + (3.65w + 5.82z)H^3/2

Where H = Upstream Head & H</=6.0'
z = slope of slot sides, z horizontal:1 vertical
w = slot invert width at base
Limits for z are 0.60>z>0.00
Limits for w are 2.0>w>0.33 ft.

You could set H at 4.0,w at 0.33 and z at 0.27 to get an overall Q then reset and subtract out the value you obtain with z at 0.0.

This equation is used pretty extensively in SW Missouri.
Good luck.