I have a cooler fitted to a R21 Renault in my GTD. 350HP SBF. The first thing is it's not needed. I also have a temp sensor in the GEBX so I can monitor lube temp. On the street, the oil in the transaxle never gets above 160-170F and that's after hours on the highway @ 70-75 MPH. On track (very hot day) for 30 min stints, I have never seen it anywhere near 200F. That data is with the pump off. That's just not hot enough to need to cool the oil.
How do I accommodate the additional oil? I just add the amount that it takes to fill the cooler and lines. The pump inlet is low, below the oil level of the GRBX so it primes fine. The oil returns to the top of the ring and pinion and 5th gear area and I have a filter in the "to pump" line and the cooler is about 3X7 inches. I think I add about another 1/2 quart. No check valves necessary.
An extra 1/2 quart isn't enough to concern yourself. On a Gal that's about 10%. Better 10% too much than 10% too little.
Skip, drain the GRBX, and put back the required spec amount. You don't need a cooler in a streetcar.