There are multiple options, but by far the most likely is what they are saying - that there is something whether a traffic shaping rule, a bad packet size setting, a VPN config, or a proxy / web handler that is mangling things.
If you know that none of those things exist, then TeamCity must have some weird configuration that interacts poorly with the settings of the VPN tunnel that another proxy does not. But that's very, very unlikely.
Can you try moving the IP address of TeamCity to match that of the proxy that works? That's a quick test that, if it gets suddenly fast answers some questions, but if it remains the same tells us nothing.