The caching period is up the the client, technically. So when we're doing template work, for instance, we'll put in a request to turn off caching altogether on our Test site. But I'm not sure what might be causing your TNEW server to keep "whitelabeling" your site. Is it possible that the template server is treating the TNEW servers differently (i.e. firewall rules, particularly adaptive firewalls) to the systems you check it from?
Not in any way I can think of. I maintain the www site and I've put nothing in place that would single out TNEW servers for any reason. Assuming it's an HTTPS request like any other, there shouldn't be anything blocking them. And again, curious why it wouldn't degrade to a cached copy rather than to whitelabel. If that's not a feature, I'd think it's a no-brainer enhancement request.
I'd definitely put in a support ticket. They may have an answer or be able to see something in their logs.
I do have an open ticket. Just putting this out there for more insight. Thanks!