Dear Vaughan,
Very interesting definition. Do you have an example of a closed category that cannot be expanded to a closed monoidal category? If there's one in your paper then my apologies for overlooking it.
Every closed category can be embedded fully faithfully into a closed monoidal category such that the closed structure is preserved; this is due to Laplaza (exact reference in my paper). However, non-monoidal closed categories do occur. I was motivated by the example of A-infinity categories, and frankly, I am not aware of any other non-trivial and non-artificial examples, but I am sure there must be some, it is just my ignorance. Best, Oleksandr -- "Dealing with failure is easy: Work hard to improve. Success is also easy to handle: You've solved the wrong problem. Work hard to improve." - Alan Perlis