The basic fact is that there is not even an exact correspondence between the English word “date” (in the present meaning) and any Italian word. One of the nearest is appuntamento, but you can have an appuntamento with a friend, with the dentist, and so on, while talking about a date involves explicitly a romantic interest, to the point that a couple might try to ascertain whether a meeting they had was, or not, a date.
In Italian you have to use a periphrasis, perhaps involving vedersi or uscire insieme (or something completely different to the same effect). There is no one-to-one correspondence between the words of different languages.
So, for our “playdate”, depending on the context it appears in, I would probably express it with a phrase with a completely different structure, about, say, «portarli a giocare con gli amichetti», «far venire gli amichetti a casa» or, often, something involving a merenda or whatever.