co-authorship of the unresponsive and careless advisor?
This is a common dilemma in academia that can quickly escalate to sorrow. My advice is short and straightforward: once the manuscript is ready from your side, politely approach your both supervisors and ask them directly about their expectations concerning authorship.
I understand the situation may feel uncomfortable and you'd prefer to make this decision relying on everyone's best judgement, but most of the times supervisors are overly touchy regarding merit and authorships. 

Thread carefully there and be frank with your superiors. Good luck!