I would go for small platters. I don't think the adults will assume they are being offered a full meal at a 2 year old's b-day party at a play place, so anything you offer will be seen as a "oh, wow! how thoughtful and generous!" Plus a sandwich, wings, steak tips, cupcakes and presumably excess pizza makes for a really nice spread.
Even if the party takes place over what is traditionally "dinner time"?
Just curious if this is an exception to the "feed your guests if the party is scheduled for meal time" rule, and if it is, then why?
Well... for one IMO its not at dinner time. In my family and my social circles 6:30/7:00 would be called "an early dinner" with actual dinner time being somewhere between 7:30 and 9:30 pm. So a 4:30 to 7:30 party would be to me, an afternoon, between meals party.
Now for little kids, an early dinner at a party makes total and complete sense to me. But not for adults.
Also though, its a child's party at a child centered place (as opposed to a party that's really as much for the adults as for the kids) with the kids being the invited guests
and the parents simply sticking around
(due in large part I suspect to the children's ages), so yeah I think the kids should be fed, but I think the adults will view it as extra, and wonderful no doubt, hospitality, not an 'of course' situation as it would be in the case of a dinner party or something.