Hi,
Generally speaking, you need to configure your browser (Chrome and others that you use for testing) to ignore expired certificate.
https://productforums.google.com/forum/#!topic/chrome/N_XYh9OrXBs might help you with this.
The bad news is that I am not sure whether or not the setting can be applied for CrossBrowser environment because in this case it is not your responsibility to launch the browser and thus I am doubting that you can provide it with the required parameters.
It also should not be possible to get assess to the 'Your connection is not private' page, so you should not be able to click the Advanced link and proceed with exception creation from test code. (Though you may check if the above is correct.)
You may also ask Support guys directly via the https://support.smartbear.com/message/?prod=TestComplete form.
Definitely, the simplest and most reliable approach is to get a new valid certificate.
