Completely potty trained means that under normal circumstances the child will tell an adult and use an appropriately sized toilet. The club has low, kid sized toilets. It means for both varieties of waste product.
It does not mean that the child never has an accident. The staff is used to the fact that this happens in new circumstances. They do have "potty breaks" in addition to the idea of "when the child expresses the need." They'll stop the activities and take the kids in their small groups, one group at a time. (The ones who aren't going continue to play).
What I mean is that within the 3-4 age group, you will have the Simba group, the Stitch group, etc., each group with 6-10 kids. At one point, the SImba group will make a potty run, etc.
They do allow you to bring a child in pull-ups, but the idea is still that they be using the toilet, and that the pull up is for safety. You will be called if the pull up is wet or soiled. Especially at night, a lot of kids wear pull ups.
If you have a couple months to go, you'll probably be fine.