I had pretty much written off one day for the drive from Vegas to LA, so perhaps a few more hours won't make that much difference!
Thats my kind of thinking. ;-)

There is nothing quite like getting there under your own steam and doing as you want. If you decide to go with it and wish to stay in the canyon then it maybe worth booking in advance. Tusayan, the nearest town to the south entrance also has accomodations. This is also the location of the Grand canyon airport and papillon helicopters do flights over the canyon if this is something you still want to do.
We took this 45 min flight and thoroughly enjoyed it, amazing views and a brilliant experience, however it may eat into your short time here.