My son used to have reactions from what seemed like just touching a milk
product, but as we went along we realized it was from kisses etc. from people
who had eaten a milk product.  The allergist said it was because the saliva
in a person's mouth would begin to break down the milk, cheese, whatever, and
then when my son was kissed it would cause a reaction.

It sounds like your child had an ingestion reaction though, or that somehow a
milk particle touched a mucous membrane, eyes, nose, mouth...  My son used to
get these too, and it didn't matter how little the particle was.  He once had
an anaphylactic reation from touching a piece of cheese accidentally, then
wiping is had well on his napkin and then using the same had to put his own
food into his mouth.  So it would stand to reason that your son could have
touched, rubbed, knocked into something in the playground that could have
caused the reaction.

Sad to say, but I think the only course of action is to avoid McDonald's with
playgrounds, and parties at Chuckie Cheese.  When we are on the road and stop
at a new McDonald's we don't even enter the building if they have a
playground.  We eat in the car, or drive to the next state rest area, etc.

Funny, but sometimes it is very rewarding to find alternative ways of dealing
with these situations.

--Sally