In short: yes, you can, just try it.
It's missing the point of the Guitar Link obviously, because that is specifically designed to work directly with a high-impedance unbalanced signal as the guitar or pedals offer. Whereas the balanced DI output is low-impedance, meaning it's stable enough to be plugged into an ordinary microphone input, but on the flip side also offers much less voltage level. So you will likely have to boost the gain in software much more than is ideal, which means a bunch of problems: not only the signal, but also any interference (hum, AD-quantisation etc.) is boosted, which may in this case be very notable (the balanced output doesn't help you, because extending it with a mono jack degenerates the connection to unbalanced again).
Likely, you get better result by feeding the guitar signal directly to both the Guitar Link and the amp, using a split (mono-Y) cable. This may cause its own hum problems due to grounding concerns, but is in this case probably the better option.
The real proper option is to use a general-purpose audio interface with mic inputs instead of the Guitar Link. I'd then record both the DI signal and the signal of a mic in front of the amp cabinet.