It allow me to use the chrome on the flashdrive but won't allow me to install anything on that chrome.
You're saying a workmates personal laptop is blocking it being installed as well? Why? Has your security team been messing around with people's personal laptops? Call the police!
Unless his laptop is using your workplace's internet/server which can also block installations. You need to do it at home. Simple as that.
Also ... you don't have to actively install it. Just go to settings, sign in to your Google account, turn on Sync and it will automatically sync/install all extensions, bookmarks, theme, cookies and saved passwords. It will be identical to the Chrome on your computer at home.
I'm waiting Vic to reply me with that CRX file by the way ---
And what if that doesn't work? I'm just offering an alternative. I've used blocked computers at work places many times in the past and installing things via my own computer and internet at home first has always worked. This includes other portable software like FTP and Image editors