This notion of breaking into the local scene is very MC. I hope Harry's long-time friends welcome them cordially.
TBH, I think HaM don't need to 'launch themselves into society' any more than W&C needed to 'break into' the so-called turnip toffs' set when they were living at Anmer Hall full time a few years ago. Harry's friends who live locally, and further afield, the friends whom Harry's been texting all these years whilst in CA, will invite H&M and vice versa.
I'm more curious to know what reception friends he and William know mutually and family will give them. Will Zara be inviting him to Gatcombe? Will Peter be inviting him to kitchen sups to meet Harriet? What about the van Straubenzees - that's got to be awkward since some of them are godfather to William's children and some are godfather to Harry's ...
Since H&M have 'done a Crawfie' so many times, I imagine most people will be avoiding them simply b/c the rich and famous crave their privacy.