Good thinking GeorgeB (I hadn't thought of that), but you can only use one of those strap tools when the serpentine belt is not installed, which means the fan and its coupling are detached from the pulley.
The belt is a hundred times easier to install without the fan in the way. It literally takes five-minutes. Once the fan is off, and the belt is on, that strap wrench does no good-it won't wrap around the pulley because the serpentine belt is now in the way. To put the belt on with the fan attached (and not have to deal with this pulley issue), is an exercise in frustration. Sure it can be done, after twenty-minutes and risking belt damage, I saw the easier route was to remove the fan and its coupling. To correctly tighten that pulley, you need that wrench.