(The Wireless Controller should stay visible in that window even if it's off, so it's not because of that.) And so, when you turn the controller back on, it's indeed a new device, because something just deleted its last "installation". If you keep the "Devices and Printers" control panel window open during this process, you will see that the Wireless Controller appears in it during installation, but then few seconds after the driver installation was successfully finished, it just vanishes.
But then, I notice that the controller is turned off (its LED is off), and when I turn it back on (by pressing PS key), Windows will again ask permission for the pairing, and starts to install the device driver again, which succeeds, but the controller is turned off again, and if you turn it back on, above cycle will repeat forever. Not sure how exactly the pairing problem for OP looked, but mine was like this on Windows 7: I do pairing on the usual way, you know, add new bluetooth device, hold Share+PS button until controller flashes, then the controller shows up, I allow the pairing, then Windows starts to install the driver automatically (from Windows Update), and even says that it succeeded.
Otherwise, the often-useful advice applies: ensure the bluetooth device drivers are up to date. It may be worth borrowing a bluetooth USB dongle and seeing if that changes the issue. Since you don't have the cash right now for the Sony dongle, you probably don't want to speculate on a new bluetooth adapter (though they should be cheaper than the Sony dongle). Myy current board with has an Intel Bluetooth controller, it worked right off the bat. So much so that on one board (Asrock FM2A88X-itx) I had to go get a $10 blue tooth usb card and use it.
I had a few major issues using a PS4 controller with Qualcomm bluetooth devices. Other have reported issues with bluetooth drivers, eg Qualcomm
Hold the PS button down to turn the controller off.(emphasis mine) Put the controller into pairing mode Many people report the controller’s Bluetooth connection can be a bit flaky on PC, depending on your Bluetooth chipset and drivers, so you may want to use a wired connection or an official wireless adapter if you experience problems.
However, note the word of caution in the next sentence: To wirelessly connect your PlayStation 4 controller with a PC without any additional hardware, you’ll need to put it in Bluetooth pairing mode.
While this could be done with a wire or dongle, HowToGeek reports that this is possible without the dongle sold by Sony:
Excellent Dualshock 4 controller windows 10 guide as always.Pairing the DS4 with Windows 7 over bluetooth