I've actually thought all of this through, here are some more details:

The factory head unit will remain in place while the wiring is
re-routed to an after market deck that will be hidden (either in a
retraction device or in a center console; either way it will be out of
sight). Two 4.25"'s mounted in stock locations with stock coverings
replaced after new speakers are installed. Same idea with the rear
6.5"'s. Everything thus far is hidden.

Ah, very good. Let me suggest, though, that you try to expand the opening
of the stock fronts so that you can fit larger speakers than 4"'s. Even
5-1/4"s will perform significantly better. Some cars are easier to do this
in than others.

The Xbox will be mounted under
the passenger seat and, if I am not satisfied with it's concealment,
also covered with carpet to match the interior of the car. There will
be no wire for the controller as it will be a wireless one (and can,
therefore, be stashed in the glove compartment). The TFT screens that
I've been pricing are only a 2" picture (measured diagonally) and can
easily be mounted on the side of the driver's side visor that cannot be
seen when put away (another idea I had was to mount the screen on the
deck if I end up going with the retraction device).

2"? Wow, that's pretty small. They have screens that are made specifically
for visor installations that measure about 4 inches diagonal.

The sub will, of
coarse, be installed in the trunk along with the amplifier.

But enough details... Does anybody know the answer to my original
question?

Yeah, it should be ok. Make sure your battery and alternator are healthy.
That's not a whole lot of power that you listed, really.