Our extended family, for many reasons, encompasses a wide range of ethnic backgrounds, classes, and financial situations. We get on well across those, our only 'hiccup'is around someone whose unpleasantness has nothing to do with money or education (and those 2 don't go together necessarily!)
First of all, I would say 'Don't borrow tomorrow's problems today'
I am hoping you like your partner (!) so it is fairly likely that his kids will be nice. Our privately educated members are outgoing, pleasant, and interested in their family's news & activities, just like most of the rest of us.
I came from a working class background, but in a solid community that taught me that I was better than no-one, and no-one was better than me, so it never occurred to me to feel inferior or superior. If that doesn't come easily to you, just try thinking about it for a bit.
Elrel has a good point, but I'd add (don't want to put identifiers) that 2 of our family who have developed a 'separate friendship' are from very different backgrounds financially, and do very different jobs - one is what I might called 'skilled working class' and the other 'academic professional'.
Also, the onus isn't on you - the baby will provide a focus of attention!