Shouldn't have any significant affect on oil pressure. Two separate issues. Plan to address the oil pressure issue separately with EITHER a different pump OR by modifying or changing the relief pressure spring.
Want an oil cooler for long trips towing. Read a story about Chevy prepping a 283 back in the 60s so it could pass German certification (forget the standard) to be driven on the Autobahn. First try was unsuccessful. They added an external oil cooler (with no other changes) and it passed.
My Euro spec 745i has on oil cooler. So does my son's Euro spec 635CSi.
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