I missed this thread in 2004, but of course it was short and I was pretty busy myself then!
Indeed the "Royal Hawaiian Golf Course" was the name for Waialae C.C. which was built as part of the hotel development down the road at Waikiki. It became a private club later, sometime between the Depression and the end of the war. It was the club who, ironically, sold off part of Raynor's oceanfront front nine for some housing and a newer hotel, the Kahala.
I have not seen Joe Mayo's name in any of the materials the club has given me, but surely someone must have come to watch over the construction after Seth Raynor's death, and it well could have been Joe Mayo. [I haven't seen Banks' name involved, either, and I doubt they would have brought him all the way from the East in 1926. Mayo was probably sold to them as the guy who supervised Raynor's work at MPCC.]