It'd be good to see the play house with your house in the background, but my first instinct is that navy blue might be fun.

Natural earth tones like the trees

With that, I think not navy blue, but dark teal. I don't think the play house has to look like the "grown up" house, in fact I think it should have a cheery look of its own, but using the colour of the trim on your house for the trim on the play house might tie them together nicely.

This is (as close as our screens can make it?) Valspar Classic Teal.