I think the exact translation of "Fix" or whether or not the factual information about government daycare capabilities was correct, is a red herring and irrelevant.
It is not the place of a total stranger to tell ANY parent what they "SHOULD" do with, for, or about their child. This person was completely out of line. I don't care if she was frustrated. Any school that has adult students, and certain/limited registration hours, is going to see some parents bringing children into the registration line. This was not a class - this was registration.
"Please take him outside, he is disrupting the line" would be ok. "Please come back another time" would be OK - the director has a right to set limits on what is allowed to happen in the office. I could even stretch to, "you need to leave", as being within the director's rights, for really out of control behavior. "Do you need a hand?" would be better. "I'll take your form, so you can take him home" would be better still. It is the director's job to control and manage the office, NOT to control or manage the OP and her kids.
You SHOULD do x,y, or Z for/to/with/about your child, because I know better than you how to parent YOUR kids. Not okay. I would definitely contact this person's supervisor, and/or the student resource office, to complain about this treatment.
PS. You weren't imagining it. That was a really rude and insulting thing to say.