Something like epoxy for painting a playhouse?
I agree an epoxy coating makes sense.  West System epoxy would pretty much do what you want.  GitRot would have better penetration, but it'd be pricey and not designed for that purpose.  West System is specifically meant to be painted on.  They even sell special foam rollers. 

It will penetrate somewhat, get underneath those wanna-be splinters, and tack them down.   It is sandable, but if it's not blended with fairing powder, it can be fairly hard to sand, i.e. The sander won't take material very quickly.  I would recommend one pass with sanding to remove burrs and odd bits.   

If you also want to level the surface you can mix epoxy with fairing filler, then trowel on the mixture.  This is much easier to sand.   

Lightly scuff-sand the epoxy, wipe off the amine blush, and you can paint it with most paints.