Hello Nathaniel,
Thank you for posting question on Microsoft Windows Forum!
Based on your query of Bluetooth not being able to reconnect with your Nintendo Switch Pro controller. Well! The suggestion here is to try pairing correctly on the controller by holding the small Sync button on the top (next to USB-C) until the lights flash, then try pairing from Windows while it is flashing. Also, making sure the controller has enough battery; very low batteries sometimes allow initial connection but not re-connect behavior.
As you mentioned the Bluetooth menu page does not pop up. Try to force open the Bluetooth Settings via URI by pressing Win + R on your keyboard and type ms-settings:bluetooth and hit Enter. Check if the Bluetooth settings page open at all.
It is also worth checking Bluetooth Support Service is running. Double-click it and ensure the Startup type is set to Automatic.
There might be a chance a policy (GPO / MDM) is hiding or blocking the Bluetooth Settings page or pair functionality. You can check by running gpedit.msc and search for Settings Page Visibility or search keywords Bluetooth. If any policy hides the Bluetooth page. If so. You might need to contact your Admin or IT department.
Hope the above information is helpful! If it helps you getting more insight into the issue, It is appreciated to consider clicking "Accept Answer". Should you have more questions, feel free to leave a message.