Won't restart till cooled

Friends boat! 2001 Evinrudes, 250 Fitch. serial number#G04886638, model #E250FPXSIF on the problem engine. It starts fine the first time and everytime after being run at lower (less than 1000) rpm's. If it is ran at higher RPM's 2000 and up for even a short time when he shuts it down it will not restart (does not turn over) for about 15 - 20 minutes. He gets no alarms while running at the higher speeds but when he tries to restart it the alarms "check engine" "Overheat" "no oil" remain lit and will not extinguish. The engines sat for about a year and a half before he started them up last week!! The water pumps were changed by a dealer last year before being stored. He hasn't had the computer codes pulled yet. Sounds like a possible overheat to me. I would suggest to him to first take temps of both banks after a run and start from there. Any help appreciated!!

Temps were taken from both banks, top and bottom. Each side had a top temp of 130 and a bottom temp of about 140. Any ideas??

Pull the codes first.

Replace the EMM.