Could `bracket` be given this linear type?

Suppose we gave Control.Exception.bracket (that is, the bracket from base) the following signature:

bracket :: : IO a -> (a -> IO b) -> (a -> IO c) %1 -> IO c	 

That is: a signature that says that the “in-between” computation will only be used once.

This, to me, makes intuitive sense: bracket doesn’t do strange things like executing the computation twice. True that it can throw exceptions, but in that case “anything goes”, we are outside normal flow of control anyway.

Would this new type for bracket be unsound, or misleading? (I’m not proposing actually changing bracket in base by the way.)