Yesterday I figured out how to get a cool wiring option on my own Strat, and then realized that this new technique also had potential for what you had requested here. So this morning I drafted up my new idea for you.
I haven't spent more time to create a color wiring diagram in software for this because A) maybe you already moved on to another wiring scheme, and B) there are two caveats to my scheme:
1. Middle Pickup active by itself (traditonal Position 3 on a Strat) is not a position on the 5 way Superswitch in this scheme, but instead, you get that option by using a Fender S1 switch to override the 5 way superswitch. I have the Master Volume pot setup with this S1 switch/pot.
2. The scheme currently requires 2 switches, compared to the "just 1 switch" you requested. Particularly, both will need 4 pole type switches So I have set the Master Tone pot up as the 2nd Fender S1 switch/pot.
I've attached the sketch to this thread, and since some forum members have told me that a diagram attached to a thread gets blurry when zoomed in, following is a link to it at my Imgur account:
https://imgur.com/gallery/V6C7yg3
With the Master Tone S1 switch not pushed (labeled as "Down" in my diagram), you have the HH configuration you requested. Both Inners and Both Outers in Positions 2 and 4 are hum-canceling.
With the Master Tone S1 switch pushed (labeled as "Up" in my diagram), you have the SSS configuration. Because the choice of which coils of the Neck and Bridge humbuckers to split in SSS mode are driven by the magnetic polarity of the Middle pickup, and you had said you wanted to use the STK4m pup whose Main coil has a South Up magnetic polarity, the two active splitcoils of the Bridge and Neck humbuckers in SSS mode are therefore the North coils. This plus the STK4M being coilsplit when the Master Volume S1 is NOT in Override mode gets you humcanceling in Positions 2 and 4. Which means Position 3 featuring Neck and Bridge North coilsplits combined is not hum-canceling. However, this is a case where you could just not even use Position 3 in SSS mode and instead just use the Master Volume S1 switch to get you Middle Pickup only (which IS hum-canceling because I set it up to be the full stack/in series in this override mode). You still have the option to use Position 3 if the venue/room you're playing in doesn't have a lot of RF noise. If you're open to one additional 2PDT switch being added to the scheme, we could use it to split one of the two humbuckers to the South coil in Position 3 temporarily to get hum-canceling in Position 3 in SSS mode.
In order to make the Neck humbucker's North coil also be the outer coil of that pickup in SSS mode (and to also have Inners and Outers in Humbucker mode be hum-canceling), I designated the Neck humbucker to have a mag flip. Which reverses the electrical polarities of the 4 conductor wires, and that is why you see Neck humbucker's Green wire being used as the Hot wire instead of Black wire.
If you were to use a neck or bridge SD Stack pickup in the middle position instead of the STK4m, those have North Up magnetic polarities, so the scheme would have to be modded so that the Humbuckers get split to the South coils instead.
And, the scheme can be tweaked so the S1 switch functionalities switch places btw the 2 pots.
Not all the ground wires are drawn, like the ones that connect the pots to the jacks, or the bridge ground wire. Also not present is a 470K resistor to use eith the Middle pickup, to knock down the brightness since the pots are 500K values.
If the caveats are ok with you, then i would be glad to spend more time and create this in some circuit design software that is in color with the correct pickup wiring code colors, cleaner wiring lines, and overall better layout.
Let me know!