I now realize that I've forgotten the design rules for the 78xx... Oh well...

Have you tried to measure the actual current draw of the WAP? Might well be that it doesn't draw more than 0.5-0.6A or so.

Also, one of the statements on that page "Change the transformer from where you take the power to the circuit to a model which can give as much current as you need from output" just didn't make sense.

Well, not if you run from a battery bank capable of delivering 100s of ampere... But if your power comes out of a transformator (110V -> say 11V), you need one that can deliver the amount of current you want. If you get a trafo that can/is designed to deliver 0.5A, you shouldn't try to draw 3A out of it...

Some pointers though - the big cap on the left is there to smooth out the ripples that you have when your power comes from a trafo and diode rectifier bridge. The more current you draw, the bigger the cap. I seem to recall a rule of thumb of 1000uF per 1A.
But running from batteries, the need is greatly reduced... I'd probably go with 470uF or so, for any ripples that might be when running the motor/alternator.

IIRC there should also be smaller ceramic caps right by the regulator, both on input and output - my mind says 100nF, but I'm far from sure...

Whatever model you get, I'd try to attach it to something that could help cool it - at 1A you'll be dumping about 8W in the regulator.

/Michael