Move Mendeley's Database to Another Computer
Your best bet would probably be an external storage device such as these on Best Buy (not an endorsement of BestBuy, I just clicked the first online retailer link). However, you should be careful when doing this. If the computer in question isn't attached to the internet for security reasons, it might not be acceptable to plug an external storage device into it without special permission.

Edit: In response to your edit, if the software absolutely requires an internet connection, then it seems impossible. If there is a loophole, you should use the loophole. However, I don't know any loopholes in that particular program (as I have never heard of it before today), and I suspect asking for such a thing would be off-topic here.