How do I drill a mortise and tenon joint so I can reinforce it with a dowel?
The simplest solution is to drill the dowel hole after you fit the joint together.

Fit the joint and make sure everything's square and true, then drill straight through the mortice and tenon in one go and insert the dowel. Chisel/plane and sand it flush and you should be good to go.

Making a mark on the outside of the mortice where you want to drill will ensure you get the location right.