(the 'only ship in quadrant' scene in TFF includes Bennett saying he needs Jim Kirk, to which Shat does the 'oh please.' to me it is played at the audience as much as the guy on the viewscreen.)
I'll have to watch that scene again; it doesn't ring a bell for me.
I did vaguely remember Bennett stating that he wanted Kirk for the mission. However, there's really no good reason why Kirk couldn't have used another ship for it, if the Enterprise was really in that bad of a shape. Which begs the question: What was the point of having the ship be malfunctioning at all? Does it really have anything to do with the overall plot of the film, aside from a few silly jokes?