From a BrE point of view "clever" is the colloquial term for the slightly more formal "intelligent," but the nuance is slight. In the UK you do hear "smart" to mean "clever" but it is AmE. An American would say "Lisa Simpson is smart" but an English person would say that she is "clever".
Of course you should be aware that "smart" is the usual BrE word to mean "well dressed" or "tidy" or "well cared for".
Clever, in AE, implies cunning, quick-witted, sharp, possibly sneaky. It's often used to denote "street smarts" as opposed to rational power.
The funny thing is that apparently clever/smart are the opposite in BrE/AmE (although of course already very close in meaning). In the UK Lisa Simpson is certainly clever, but because she is a 'nerd' she is not smart. If I understand our American friends correctly exactly the opposite is true there. She is certainly smart but not clever.
I know this hasn't exactly answered your question! At the end of the day, I think you can rest assured that you can pretty much use these terms interchangeably with only the slightest of difference. As a BrE speaker it would seem more natural to use clever in your example above as it has the connotation of problem solving........and yes sometimes 'the smart thing to do' refers to 'the sensible thing,' however again I think this has more to do with AmE / BrE differences than grammatical correctness. Maybe a native AmE speaker could help me out here?