I wonder if the two are genuinely playing, or if the dog is playing and the fawn is desperatly trying to defend itself against the monstrous blonde beast.
The trick is to put them in a drawstring sack with just the head sticking out. Just make sure it's heavy canvas or burlap so they can't claw through it.
It sounds cruel, but there's actually less chance of them hurting themselves (or you!) that way, and I find they stay calmer.