You can gather the Sitecore TDS License expiration date for an active installation, looking at the ExpireDate subkey of the Computer\HKEY_CURRENT_USER\SOFTWARE\HedgehogDevelopment\Sitecore Visual Studio Integration 2.0\ActivatedLicense15 key in the registry of the machine where TDS is installed.
Note that this is the registry key for TDS 126.96.36.199. I don't ...
As far as I know, there is no easy way to find it.
The only 2 ways I know are:
if you have an invoice for the license, there is information about the period when the license is valid
if you requested new license via email, you should have the information about the period when it's valid in the response from the sales.
I had the same problem and after the TDS installation finished and before the VSIX installation I ended all msiexec processes and after that I did the VSIX installation. With these steps I was able to install TDS and the extension on VS2019.