Once again, thank you very much chaps and to those who have sent PMs. This is very useful stuff, although perhaps is not leading towards the conclusion I hoped. I have written to the embassy to get the latest info in writing from the horse's mouth. I'll let you know what they say.
I suspect that they will tell me that an M-1 visa is mandatory for the flying I wished to do. I am rather put out by the fact that the school I was dealing with is on the AFSP approved list despite being Part 61 and not SEVP accredited. Moreover, the school knew my circumstances and assisted with my AFSP clearance. As far as I can see, this setup absolutely guarantees that students who apply to some AFSP approved schools will be wasting their time and money as there is no possibility of being issued an M-1 visa for attendance at that school. The fact that the school was complicit in this state of affairs (probably through ignorance) leaves me very annoyed.
I will absolutely not take any chances regarding improper visa status as I hope to graduate from military to airline pilot at some point in the future. Being banned from the USA would not be a good start.