The friendship issue is quite difficult at the age your son is at. Other kids are very tuned in to the fact that someone is a bit 'different' but not emotionally mature enough to empathise with their situation, so they tend to avoid or bully - horrible! This has improved dramatically as my son has got older and has been considerably better since he has been at grammar school. We have found that the key to this is common interests. Over the years we have gone from Beyblades to Pokemon, Yu gi oh and currently it's Magic the Gathering. Boys don't "chat" the way that girls do so conversations tend to revolve around their interests and being able to join in that helps enormously. I know you're concerned about things going astray - my son loses everything
but I've given up worrying and just tend to replace things now (unless they're really expensive).
I can understand why you're reluctant to tell the school about your concerns as I was in exactly that situation when my son took the entrance tests to his prep school. At the time we didn't have the Asperger's diagnosis, just a diagnosis of dyspraxia and I chose not to tell. However, a girl in my daughter's year with some undiagnosed special needs did tell the school and they were quite happy to take her. The school very quickly noticed that my son had difficulties and they were the ones who persuaded me to seek some help. I do think it would probably be better for your son, and easier for the school, if you were honest with them from the outset. You don't have to mention Asperger's, just explain some of your concerns to them. It could create a bad atmosphere if they later find out that you didn't make them aware of all the circumstances.
A good school will not turn your son away because of this and it sounds as though he can cope quite well in a mainstream school. My experience has been that on the whole the teachers are very understanding and helpful and having the diagnosis has been really beneficial for my son. I can't remember where this quote is from but it's very true that the diagnosis "is not a label, but a signpost".
If his singing gets too much for you I would willing swap him with my daughter - she also sings 24/7 but with an ear piercing tone and a drawling American accent