Hi No_Face,
I'd like to clarify your situation firstly. You want to re-run the Microsoft Sign-In Assistant to sign in the desktop with another Office 365 account, so that you can re-active Office Professional Plus subscription license. Am I right?

To re-run the Microsoft Sign-In Assistant, please go to Manually install Office 365 desktop updates, download and re-run it here.

Meanwhile, to refresh the Office Professional Plus license on the client, we need to  reactivate it by using Osaui.exe. Please do the following steps:
1).Address the Osaui.exe at the laptop. It is located in the Office14 folder, you may perform a local file search to find it.
2).Make sure that you are connected to the Internet.
3).At the command prompt, type the following command, and then press ENTER:
   OSAUI.exe /K
4).Type the Microsoft Online Services ID and password provided by your administrator for Office Professional Plus.

For detailed guides, please refer to Reactivate subscription license by using Osaui.exe

If you have any other questions, please feel free to post them in the forum.

Thanks,
Evan Zhang