We made the choice to limit children invited to immediate family as our friends and coworkers all have multiple children and we would've blown our budget and venue max quickly. We made sure to clearly address the outer and inner envelopes with the names of those invited but people are RSVPing and adding children. Do we wait and see if more people do this and then make phone calls? I do feel bad because our response cards include a line for children's entrée for 12 and under, but we needed to include it because some out of town family we don't have other contact information on have children around the cut off age to and adult entrée.
Answers
Also, you should never have put a line for 'children's entrees' on the RSVP card. That's the source of confusion for your guests, and rightfully so. You said you feel badly, but you're about to cause a boat load of hurt feelings.
I know you can't do anything about this now, but for lurkers, I actually intentionally left our kids meal option off our RSVPs because we were limiting the kids to just immediate family, and wp kids (which covered all OOT), and I then added a handwritten note with their invitations about the kids meal option to those specific parents. We still had one couple without kids invited write in a kids meal "if they are an option", so it's not foolproof.