Hey eddie,
I’m a big believer in the 4-switch and along with Grumpy Bill the fuller tone is in this switch, use when needed.
Don’t have a ASAT Classic with a 4-way but have modded 4 of my bunch and a mix of pickups among them. One is close to what yer looking at.
Guitars with 4-way include: 88’ ASAT, parts ASAT Z2, ASAT Deluxe with push/pull for both pickups and a 4-way, and I built essentially a fender Baja which has the 4-way switch Broadcaster bridge and custom shop twisted tele neck. This is close to the Classic.
For a low cost mod, between the switch and install, even cheaper if you do the switch install, I think you will find some tones you like.
Let us know what you do. If you do, it tell us about it.