Will the $100 big brake kit work with 13" wheels?

The short answer is maybe, but not likely. I'm sure it depends on the 13" wheels.

The center line to the high point of the caliper is 5 7/8" you would need 1/4" clearance. So you'd need an inner diameter of 12-1/4" at the inside of the wheel. The wheel profile is also not flat, it tapers from larger on the inside of the car to smaller towards the spokes. The tightest clearance on my 14" wheels with 30mm offset is by the spokes. That outside part of the caliper also has a smaller distance from center line (5-5/8") than towards the inside of the calipers where the cooling/reinforcing fins and fittings are.  I have over 1/2" at the outside point of the caliper ( by the spokes), but with the balancing weights, there is less than 1/2".  So theoretically it is possible that the big brake kit will work with 13" wheels but it would be really tight.  Caveat Emptor.

If you wanted to test it you could buy 1 rotor and 1 caliper, slide the caliper over the rotor, and bolt up your wheel with out it being mounted on the hub...allow for 1/16" between the rotor and the carrier.  If it doesn't fit, return the parts for a refund.  You may even be able to take a loose wheel to a local auto parts store and have them pull the parts and see if they provide the clearance for your particular wheels and offset.