Given the solution for the classic Bluetooth devices in Mike Petrichenko's answer, I did some research on how to use WinRT and raw PowerShell scripting to do the for Bluetooth LE devices.

You can find my full solution on GitHub Gists, but generally it heavily revolves around using the Ben N's solution on calling UWP API (and so WinRT API) within a PowerShell script.

That solution, though, unfortunately does not work for PowerShell 7+ due to "removal of builtin support in dotnet".