If you truly are willing to pay for only 15, then invite only 15! I know folks say "invite extra because some will say no" but if you do -- be totally prepared to pay for all 18 kids without any regrets. The one time you figure on at least three kids saying no will be the one time that everyone says yes.
Does the facility where the party is allow you to give them a final head count by, say, three days before the party, so you can really know what you'll pay for? If that's the case, you will have to be very firm with people about "the facility needs a final head count on (date) so RSVPs are essential before (date)."
The invitations should say clearly something like this: "Sally is invited to Susie's party at.... Invited kids only, please." If a parent turns up expecting you to accommodate a sibling as a party guest, just be honest then and there: "I'm sorry, but we've got a party package arrangement with the facility and the invited kids are the only ones we can accommodate. If you need to stay nearby to pick up Susie and need someplace to go with Sibling, you could take Sibling to...." and name a nearby place like a kid-friendly restaurant or mall with an indoor play space,etc. Think ahead and have those suggestions ready to go. Don't like telling folks to basically take a hike with the sibling? Assign your husband or a trusted friend or "auntie" to that kind of door duty. Mostly parents do realize that parties at facilities that aren't someone's home do cost money and do limit numbers, so I hope you won't have to give anyone that speech!