I am using conventional tandem seals on a hig pressure ethane pipeline application. The pump is a multistage between bearing and the only flush take-off I have available is off the discharge end. I have about 800 psig and I am not wanting to take full pressure drop across a single orifice.

I am considering taking a .156" orifice to approximately half the pressure at about 16 GPM. I will then be branching off a tee with individual supplies to each end of the pump. Do I need to size these orfice plates for half the flow? I'm only flowing 16 GPM past the orifice so I should design each orifice for 8 GPM correct?

800 PSI is not that high if you are using a High Duty seal. I assume that the seals have been in service successfully already?  What you say is correct however this assumes both ends are identical. That is the seal leakage rates and throat bush clearances are same.  Obviously if one seal leaks more than the other or bush clearances open then you will have preferential flow.
Using a conventional seal, on this application you may experience high face wear due to the high VP of ethane.  Trust your experience is good!

Cheers