Alternate DNS server not resolving after first one fails to resolve

Primary and secondary DNS server are expected to give the same result. It seems that you have setup your own DNS server want the OS to use your DNS server for resolving your lab names/IPs and use the company DNS server to resolve all other names/IPs.

This is not the way it works. If your PC asks your DNS server for the IP of a company server and your DNS server answers that the name is unknown, then there is a result and there is no need to ask another DNS server. Also if your PC asks the company DNS server for a lab name, then it will answer that it does not know the name.

You should configure your DNS server to forward all requests for domains other than yours to the company DNS server. In an ideal case the company DNS server should be configured to know your sub domain and forward all requests for your domain to your DNS server.

You should use your DNS server as primary DNS server. If you can configure the forward zone in the company DNS you can use that DNS server as secondary, otherwise you should not configure a secondary DNS server.

And as it seems that you are not knowing how DNS works, you should ask you company admins to assist you.