There are two things going on here.
First: the "Could not establish trust relationship" error you are getting is because the certificate SWIS uses for HTTPS is intentionally not a valid certificate. It is self-signed rather than issued from a proper certificate authority like Verisign, and its common name is CN=SolarWinds-Orion instead of whatever the FQDN of your server is. You will need to bypass certificate validation with a line like:
ServicePointManager.ServerCertificateValidationCallback = (sender, certificate, chain, errors) => true;
In this mode, HTTPS provides confidentiality and replay protection, but not host authentication.
Second: unfortunately at this time SWIS does not support authenticating Windows accounts over HTTPS. This is something we are looking at fixing in a future release. Until then, you will need to use non-Windows Orion accounts for this API.