What you want is DevCon from Microsoft. You can't download it by itself, but you can get it by following these instructions:
This page has a bunch of examples on how to use it:
You can find the Hardware Id in Device Manager under the Details tab in Properties for the device. If the device has multiple Ids, I suggest trying the shortest one first. You'll also want to look at Inf name. This will tell you the name of the install file to plug into DevCon. The path is not listed, but since it's a built-in driver, it should be in C:\Windows\inf\
Before you start using DevCon, you will need to uninstall the driver you got working, after you get the info, and then rescan for new devices. This should put the computer back in the state like you just imaged it. I've only used DevCon a couple of times, but I suggest using the shortest Id, if there multiple, and trying the update argument first. If that doesn't work, then try the install argument. If that doesn't work, try update with the next Id. It should eventually install. That will give you your command to run. Just add it to SetupComplete.cmd or a GPO along with DevCon and you should be good to go.