This ALMOST works perfectly, so I'm already incredibly thankful for this.
It will open up Dolphin and DS4Windows successfully. The only issue is that DS4Windows.exe sticks around even after Dolphin closes.
This is how I've got the code set up to point to my DS4Windows directory:
Run, "F:\Essential Programs\DS4Windows\DS4Windows.exe",,Min
Esc::
{
Process, Close, DS4Windows.exe
WinClose, ahk_exe Dolphin.exe
}
I've also got the mute button on the PS5 controller hotkeyed to shut off Dolphin, if that matters.