I think you can fit a 2.3 tire for the max on a giraffe, I will have to check once I get home.
Take it to your LBS and ask them to find the fattest tire they can fit on it, they probaby wouldnt mind swapping out about 4 tires to find the right size.
At unicon some guys had a custom trials giraffe with belt drive, a hand made frame and a trials 19" rim with the CC 2.5" tyre, it was fantastic. I think you could probably go a little larger in the TX, however if you are considering doing trials on it they do bend easily, my friend bent the axle on his doing nothing larger than dropping curbs. I have one aswell and the frame flexes massively if you try to hop on it. The limitation with the tyre is the width rather than the height so you could get something which was knobbly but not particularly wide if you want more grip for rough ground.
That's beautiful, but it seems somehow different from a standard TX. Must be a pain to ride, I've ridden that tyre briefly on a normal uni and found it pretty difficult to turn on.
i don't know about the size of the tire ... i suppose it would depend on the size of the frame. if anyone gets into trials girraffe, send me a link to a video so i can watch. it sounds really hard to do!!! good luck.