I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes).
Then the question: Is there any (nontrivial) category of spaces that is cartesian-closed and has the Whitehead property? If not, is there some counterexample we can construct to show that these properties are mutually exclusive?
I am familiar with the category of m-cofibrant spaces, but it is not cartesian closed (even though there is the weaker result allowing us to take loop spaces etc.).