Issue is you using the Tomei pump and the volume. Use a N1 and you'll be fine.
If stock sump'd GTR's (300-350rwkw) on semi comps have run 1.20s @ Sandown with nothing more than a litre overfill, N1 pumps & restrictors - and not spin bearings... Good enough for what you want.
Obviously an oil cooler is a good move, and just keep the rev's sane for a stock tolerance motor. Should be fine, plenty of others tracking with 250-300rwkw for a few years without issues.
The latest "fad" seems to be external oil drains. You would've hardly seen anyone using one before about 18 months ago, so just ignore that as well as they are just a bandaid for poor oil control as part of a build (restrictors & enlarged returns using higher volume pumps).
Plus you aren't from NSW where legendary Sump Baffle Syndrome runs rife, so you've got that advantage on your side as well.