What bandpass filter design will yield the shortest impulse response?
The Butterworth filter's frequency response is the result of specific formulas and its characteristic is the flat passband frequency response. Consequently, if the coefficients of the IIR filter are modified in any way, the filter might not maintain the "Butterworth" characteristics.

In addition to the responses by "Hilmar" and "Jason R", maybe you could treat this as an optimisation problem of a fitness function that captures your specifications. 

You could for example start with some design (e.g. a Butterworth filter) and then use an optimisation technique to move the zeros and poles about their locations (or modifying the design by adding / removing poles and zeros) trying to achieve your specifications (a sharper time-domain roll-off maintaining bandwidth and filter order). 

Along this line, a large amount of work has been performed on designing filters with Genetic Algorithms (and here) and Simulated Annealing (and here) which you might find useful.