Gotcha. A wireless extender. That's sort of a weird inclusion. I hadn't paid close enough attention to unboxings to know Logitech was packaging that with any of their wireless mice.
So you've been plugging your bluetooth adapter for the controller into that extender cable? No reason why not, I guess, if you want to get the transmitted closer to your controller, but I'd only do that if you were having issues plugging the USB transmitter directly into the motherboard. But yeah, my recommended troubleshoots are the same whether you do or you don't.
(1) Disable the onboard motherboard bluetooth.
(2) Download and use DS4Windows if your'e still having issues.
I had not, but I'm unclear. What exactly is it combining that isn't already combined?