To be perfectly honest it's a terrible idea, don't do it. I appreciate the dilemma you have, but I highly recommend just not doing that.
I've never personally attempted to do what you're trying to achieve, but I'm fairly confident you can't install SQL Express on a desktop but have the instance directory including all the database files on a totally different machine. Even if it was possible to do that, it's still a bad idea. If the network was to drop, which they do regularly, you could end up in a big mess with that kind of setup. Also, Vault server transactions via the ADMS would be redirected over the network to a remote database and filestore, which again I'm sure you can't do with Vault Basic, but even so that would be terribly slow and unreliable.
Not a helpful answer. So if I was in your shoes, my first goal would be to do everything in my power to get a new server signed off. You don't need to have a high end server for Vault Basic in an environment which is relatively minimal load. A very basic rack or tower server would be fine, Dell do them for around the same or less than the price of a CAD workstation. Most engineering companies' management understand the importance of their design data, so dismissing the best and safest options of hosting & backing up that data is a dangerous game to play, complacent, and asking for a disaster. Having a dedicated Vault server should be right up their with the biggest and highest of priorities in an engineering business.
If that's still not going to happen, your only real option is to install the full Vault Basic solution on the desktop PC. Just be aware that if you ever upgraded to Vault to a 'paid for' version, they're not supported on desktops as of 2015. Make sure you wipe the desktop PC and install a clean operating system on it, try to avoid running another programs on that PC, and make sure there's no troublesome firewall or anti-virus programs on it which would cause chaos.
It's really not that unsafe to do this as long as you actively manage and monitor the PC. Use a script to perform Vault Console backups every night, that will backup the full Vault to a folder... then either manually copy or script copy the backup folder to the domain controller after the backup finishes ensuring the folder gets copied to a location which is backed up to tape.
If this is all sounding way out of your comfort zone, I highly recommend contacting a local Vault Specialised Autodesk reseller and ask for their help. I've done countless jobs of this nature, it should take less than half a day to install Vault Basic and set up a backup routine, providing you don't require any data loading into the new Vault. Then you have peace of mind and a solid reliable company for ongoing support.
Hope that helps.