Put a larger cooler on it, duct it well, and be done with it.
With my RCR T70, which saw mainly track use and a lot of it, we started out with oil temps at 250-275F for the first couple of outings. After we moved the oil cooler around, and made ducting so it was getting cool air in and through it, the oil temps dropped right where they should be, 220-225F. We also insulated the cooler from the radiant heat of the engine. That car would run 225F even on 95-100F days with session lengths as long as one hour. We experienced the same problem with the transaxle cooler, temps there were hitting the same or similar, but we moved that cooler as well, made sure it was getting cold air and had no problem.
Oil cooler placement and ducting, even ducts after the cooler, are key. We had the same experience on the pair of SCCA Mustangs we built. At first oil temps kept getting a bit higher than we liked, but we moved things a little, made sure air was getting into the cooler and out of it, now we run 215F all day, in any temps, in 1.5 hour races. The AN line adapter we use for the cooler does have a thermostat in it, the Lola did not.