Hello Alex,
Here's a few thoughts. First, let's verify that the switch you bought has 3 distinct "on" states. If that switch is good, this will work in the end. In state 1 everything goes to the "left". State 3 everything goes to the "right". State 2 half of the switches are going left and half are going right. It looks like from your original post, State 2 has the 3rd and 4th rows switched from my original schematic which shouldn't be a big deal as long as we switch up the schematic accordingly. Make sure this is absolutely the case. And make sure that everything is on-on-on. Some switches may throw in opens or be momentary. BTW the link you had for the datasheet didn't work for me.
After that I would verify that when you removed the caps from the schematic that everything was removed and connected appropriately. I know you didn't have the caps in there but you had to remove them from my original schematic. I have redrawn a schematic below that removes the caps and switches the position of the third and fourth switches.
The next thing I would do is make sure that both of the pup green wires (neck and bridge) are grounded. In fact you should check all of the ground connections in your wiring to make sure they are all good. It is easy when doing rework to forget grounds. Electrons need a path out and a path back. Some folks aren't familiar with the ground symbols used in schematics. The ground symbol makes it easier to read rather than connecting everything to a ground point in the drawing. Be sure that every ground symbol is connected to ground in your bass. I usually use the backs of the pots as my ground points which is the way things come from the factory.
Also, after verifying all the wiring against the diagram below, try things one pup at a time? Does the neck give you three distinct state? And then the bridge? Parallel and SC can be similar if you are not playing loud. You probably get some SC noise though when going from parallel to SC when soloing a pup.
If you continue to have issues, we probably will want to go thru the ground connection checklist which is in the Supermod document. All of the pots need to be connected to each other. Battery ground. Jack ground. Pup cavity ground. 4-way switch ground. Volume pot grounds. Treble pot ground. etc. And detailed pictures of the 4-way switch, pots, etc, may be in order.
https://drive.google.com/file/d/1TlVRN7q0mmacBDIoxm20M5zIp0rPv6oi/view?usp=drive_link