I will not hurt to drive it as is.
Are you sure the pulley is too small? Remember the shaft and pulley are press-fit, so there is no taper...the pulley should not be able to slide on by hand.
I used the pulley boys puller and pulley, so I am not sure of the difference between it and the ASP one, though on mine they recommended liberally coating the shaft and pulley with never-sieze before installation. This is to help with the installation since it is a press-fit.
As well you could try to heat the new pulley on a hot plate before installing it(remember to use gloves when handling), as the heat will expand the pulley, and give you more clearance.
Good luck!