I think my son is one of only a few who does not invite all his classmates to his birthday parties. Many either go without a party, or wait until they can afford all 20 kids to come. Personally I limit my kids to only their closest friends, (for example maybe 6 if we are having it at home over lunchtime, and 2 if he wants them to come for dinner and a DVD) and for a playcentre I would only invite the few that he is good friends with. However this is the norm for the older kids who in general don't want every kid they know to come to their birthday and would rather limit the numbers.

So for your daughter, if you really want a playcentre party, then only invite those you and she are closest to out of mothers group. Sooner or later all the mums will start doing the same and limiting party numbers, and I'm sure if you don't make a huge deal of it they won't mind.