Thanks for the tip Racer65. I bought the aftermarket pump (reluctantly since it's not Toyota) and installed it.
It was a bit of a tight fit but it did work as you mentioned. Here's a picture showing the wiring. The striped wire is the positve.
I thought the same thing but started having some little bubbles at the curve on my dash pad, so I got the SOR dash cover a few years ago.
It is tolerable to cover your dash with a rug but not your head IMO!