What is a stable frosting for a cake that will be in warm temperatures at an outside party?
Here is an interesting entry from King Arthur Flour, who should not have a major stake in the game.  They hit a bunch of varieties with a hair drier to see how they held up.  Their test though was only on holding up to the heat, not safety though.  One could suggest using pasteurized eggs as an option though to get around safety concerns.

Quick summary, they liked Italian Buttercream among the buttercreams they tested.  Shortening over butter helps, but I am in the group who really dislikes the taste of "buttercream" made with shortening over butter, but would go by taste on that.  Of the ones they tested though, cream cheese frosting held up the best, so if the cake is one that would match with cream cheese it might be a good choice.

Fondant is another option which may hold up well to heat though personally, I find it for looks only, not taste.