Im pretty new to all this, not sure it will work (as I have not tested it) but there are some usb devices that can plug into your pi zero that allows for ethernet connection. This would allow you to hook up your pi zero to the internet via ethernet and thus install all the required extentions to get your GUI working. "Three Port USB Hub with Ethernet (micro B) - by Pimoroni" is one that 'reviews' say works. They don't state what kind of OS they are using at the time of 'it working', and perhaps TinyCore does not have the drivers required to make this work, but its an idea.
There are other usb splitters with ethernet ports that 'users' say work with the pi zero (without wifi), so you may not be limited to the one stated above.
Honestly I am planning to buy one and test it out at some point, but won't be for some time, so if you get it to work, then let me know!