Is there a Penrose diagram for two black holes near each other. Perhaps they are colliding or circling each other? Or can this method only describe a single black hole.
The first step in drawing a Penrose diagram is to make an $n$-dimensional section or projection, where usually, but not always, $n = 2$. We want to take advantage of any symmetries, such as rotational symmetry, so that the final result will be informative, be representative of the whole spacetime, and accurately depict causal relationships in the original spacetime. Your spacetime with two black holes has a low degree of symmetry so this will not work for $n=2$. At this step we also need to make sure that lightlike geodesics in the original space correspond properly to lightlike geodesics in the submanifold.
The next step would be to apply a conformal transformation in order to make the diagram compact. If you want the diagram to be flat, then this requires the $n$-dimensional slice or projection to be conformally flat. All two-dimensional manifolds are conformally flat, but in your example we will have $n>2$, and I don't think you're going to get conformal flatness.
So no, I don't think you can draw a Penrose diagram for this spacetime.