geoff, i don't have any photo documentation, but was rather taking a stab based on the mansion Scott pointed out. Therefore, I'll take your word for it. I think the LIE wasn't built until the 50s, so clubs began to be forced off their properties in the 40s.
How about the old Meadow Brook Club? The course was forced to move in the 50s, so presumably it would still be functioning in 1938. It would be helpful to know if the LIE moves through this property today, as it would narrow down the number of potential courses.