Doktor Avalanche
Ultimately you need confirmation that it is unsigned from somewhere.
The somewhere could be Tascam/Teac support.
The fact that you have the driver installed on one Win7 Pro machine but an unsigned driver issue with the other brings up the question of whether the laptop is 32 bit and the desktop 64 bit. Driver signing in Win 7 is only enforced on 64 bit versions. .
There is a driver for the unit on the Tascam website dated 2015-7-9 v101 that you should try if you have not already.
http://tascam.com/product/us-16x08/downloads/ There is of course no way of knowing if that is really the Tascam website and no verification hash is present so you could still be installing a virus if an evil genius has hacked the site. That is the risk you take with all software from whatever source. There was some discussion on the Tascam forums that an earlier driver version that was released had serious problems and that they were asking distributors to hold the product until it had been fixed.