Re-Connect Microsoft Bluetooth Surface Keyboard

I'm running Windows 10 2004 (OS Build 19041.264). I have a Gigabyte Aorus Master X570 motherboard with built-in Bluetooth and Antenna.
I have a Bluetooth Surface Keyboard which I like, but it sometimes disconnects itself.
I can re-connect it via Settings->Devices, but in order to do so, I have to un-pair it, then pair it again and connect, which is a pain.
In the 'Notifications' area of the desktop, there are two options relating to Bluetooth, The first, which has a Bluetooth Symbol on it, just switches Bluetooth on and off. The second, labelled 'Connect' does not list my keyboard, its functionality being restricted to 'wireless display and audio devices'. It has an option (at the bottom) to 'Find Other Types of Devices', which just takes me into Settings -> Devices where I have to un-pair, the pair and connect, just the same as I describe above.
Does anyone know if there is any way that I can easily re-connect a paired keyboard device which is switched on, without having to un-pair and re-pair first?

As you suggested I have downloaded the Find_ANY_Problem_Devices.bat and have run it as administrator.
This is the file it created: Prob_Devices.txt

Just in case this attachment hasn't worked, there is just one fault found, and it DOES refer to my keyboard. It clearly indicates that there is a driver missing. So, many, many thanks for that and I have copied the text below.
What are your recommendations for getting hold of the suggested driver? ======= =============== ==================================================================================================== ================== Device PNP Device ID Error Code ======= =============== ==================================================================================================== ================== Standard PS/2 Keyboard ACPI\PNP0303\0 This device is not present, is not working properly, or does not have all its drivers installed.

Hi Paul, Thanks for that.
Unfortunately, under "Keyboards" there is no entry for "Standard PS/2 Keyboard". Just two identical entries for "HID Keyboard Device", and when I perform a "Search Automatically for driver...", for these, the search takes a few seconds and returns telling me that the latest drivers are installed. This, by the way, is a driver dating to June 2006.
Under Bluetooth, there is an entry "Surface Keyboard" (which is what mine is), and when I try to update the driver (which is the same one as I refer to above), I get the same message.
I have done a search for ACPI\PNP0303\0 on the net and can find links to sites that offer drivers...I'm a bit wary, though. Forexample: semantic.gs/acpi_pnp0303_driver_v2_download or https://sciencespaces.com/search/ACPI\PNP0303
Have you any further thoughts that might help?

Thanks for both of these Paul. I've already been there, s it happens. WU is useless, and the Mouse and Keyboard Centre App just shows my Surface Mouse, not my keyboard. When I look at its list of keyboards, mine isn't there! Hey Ho!