How do I use Windows Server Backup to only backup files of a certain type?
It looks like from the GUI, what you can only do is specifying file types to exclude, but not the other way around. wbadmin and Windows Server Backup cmdlets, its command line and Powershell counterparts, however, allow you to do that.
For wbadmin, see
-include parameters of wbadmin start backup. For Power Shell, take a look at New-WBFileSpec cmdlet. Both of them allows you to specify inclusion pattern using wildcards. wbadmin will probably get your job done faster though. This article provides samples of using both approach.
Note that these tools are just alternative interfaces to Windows Server Backup. You can start backup from wbadmin and still monitor progress from GUI.