Dear Mr. Keene,
That is usually a "mismatch" between the certificate and Windows. There might be several reasons for such error, we have to investigate what's going on.
First thing is to leave TMS Sparkle out of it. Instead of using TMSHttpConfig tool, try to bind the certificate using native Windows netsh command-line tool. This documentation topic explains how to do it:
My guess is that you will get the same error, thus ruling out TMS Sparkle and then we can focus on the certificate itself.
It's good that you have the certificate installed in your local computer, but it's already strange that such certificate appears in "Trusted Root Certification Authorities". That's not usual and I have no idea why such certificate is there. Are you sure it's your certificate appearing there, or a root certificate?
That might be one indication, I'd suggest you remove/uninstall your certificate from all such places, and then install/import it in your Personal\Certificates store. If possible, please provide the steps you used to install the certificate.
It happened before with customers, for example:
Other customers, from e-mail, reported this:
We found out it was because we were missing the private key that belonged to the certificate.
More references that helped in previous support requests:
https://www.serverknowledge.net/web-server/a-specified-logon-session-does-not-exist/