@7zlzpma4s8m4 No, this isn't possible. The EA App doesn't allow you to choose the install location, although there are sometimes workarounds for the issue. But you'd need the App installed somewhere in order to play. So if you couldn't get it installed on the external drive, you'd be out of luck.
Then, even if you're technically not downloading any game files, the EA App won't run Sims 4 without at least verifying the install, and that involves creating registry entries and some other hidden files. So unless the permissions on your work computer are very sloppily done, I doubt you'd be able to get Sims 4 working.
What you might be able to do, if your USB is fast enough, is install Windows on the USB and then boot into that while it's connected to your work PC. But this might not work either, depending on the security settings on the computer.
To be honest, it sounds like your work really doesn't want its systems used for any purpose other than work, and it's probably worth respecting that.