Your DNS works just fine. What you want to do is to add a search domain so that any single hostname is automatically completed with the search domain, and the outgoing query is for a FQDN, which we already know to be working on your network.
If your local and remote LANs are called, for instance, campus2.mycollege.edu and campus1.mycollege.edu, then add the second search domain, campus1.mycollege.edu, to your Windows pc, as follows:
- Control Panel → Network and Internet → Network and Sharing Center.
- Click on Change Adapter Settings on the left side.
- Right-click on your network adapter (normally "Local Area Connection" or "Wireless Network Connection") and select Properties.
- Select Internet Protocol version 4 (TCP/IPv4) and click on the Properties button.
- On the General tab, click on the Advanced… button.
- In the Append this DNS suffixes list, add campus1.mycollege.edu.
You may now try it, to see whether this works (it very much should).
This is good for testing. But how do you implement this in an automatic way? You will have either to change the configuration of your VPN server so that an option describing the new search domain is pushed to all users, or you change your local DNS so that this new, extra search domain is added to the existing one. Both are implementation-specific, but for instance by Googling I found this for enabling search domains through the VPN, which should be relevant to you.