Hi op! This post resonated with me because we moved to a similar catchment area around six years ago, and my dc started to attend a school that was mainly middle class with mostly middle class friends.
To be completely honest, i did feel like you did in the beginning. I went over to a couple of their houses (think four/five bedroom Edwardian houses with pianos), and initially I did feel a little embarrassed about our situation as not only were we in a council two bed flat, we were also quite skint ! Dc would come home and tell us about their friends holidays and it was impossible not to compare.
It would have been really easy to hide away and in dc nursery year I did. Until at the end of the year her teacher sat me down in parents evening and said how lovely dd was but how she hadn't found that special friend she plays with, and how she is quite often alone in the playground. The teacher advised me to allow opportunities to socialise outside of school as many of the children at the school did via playdates and clubs and that cemented their friendships at school. I drove home in tears with images of dd lonely and sad in the playground ! (I'm sure it wasn't that bad in hindsight, but at the time it felt like the end of the world!).
So I started to invite her friends over and also to accept invitations for dd to visit friends too. I'm going to be honest, a couple of the children that came did ask where the rest of the house was, where the downstairs was etc
obviously though, they were just so used to living in larger homes that naturally they were a bit baffled at the prospect of a small flat ! The parents were absolutely lovely though. Never ever had the slightest of snubs/snobbish comments or anything of the sort. I've built some very good friendships with lots of the mums and our house situation just isn't an issue any more !
Our situation has improved now and we are moving into a 'proper' house and I'm so glad that we established those relationships and friendships with the children and parents at the very beginning as doing it now after moving into a home that we could be proud of would be just wierd and .. yeah, really weird basically! 
So the point of all that extremely long post is to just go for it! Don't hold back, let your dc form good strong friendships out of school and do not give a shit about the size/whatever of your home. Those that will matter won't care and those that care won't matter ! 