Interesting problem. Is the IP that is allowed to talk to the HQ Server and the IP(s) that is (are) not all on the same subnet? If not, would expect a static route to HQ Server subnet for the "correct" interface to work in addressing the problem.
Bottom-line, I expect the OS is going to determine which interface to use heading outbound, and HQ is going to let it. HQ has properties to set the IP for listening for incoming connections, but do not think there are configuration options to guarantee the outbound IP.