Yes I would. You are right about it being best to avoid digging wood and sawdust into soil, but on top is fine.
Some of the urine may leach out and be lost, so composting would be better, but gardening is all about making the most of what we have and I think it is a good opportunity to feed and protect your soil, and keep weeds down too.
I just made a hotbed (probably warm bed in fact) out of similar fresh woody manure. Green Books have brought out a nice work by Jack First about it.