I contacted Culham College's archives earlier this year. They said that I can go and look at them and that there are relevant records there. However, I have to go to Oxford to look at them and book in advance. I didn't realise though that he must have been one of its first students.
I found one or two schoolmasters in later records who seemed to fit the bill but nothing conclusive. The problem is that Morgan is such a common name it's hard to find the right people.