Just for completeness, I'll post a kind of negative answer here. My point is that I would advise against extrusion.
The kitchen equivalent of a caulking gun is a syringe-like device that's intended as a reusable replacement for pastry bags. The more widespread version is made from not-so-stable plastic and is intended for cake decoration. It works with soft substances like buttercream, and if you pack it full of goat cheese, I expect it to either not expel anything, or break. The sturdier version, which is used to press a log of cookie dough with shaped edges, may be able to extrude goat cheese - or maybe not, since the ones which can be fit with a small diameter nozzle are more likely to be intended as double-duty, and it's possible that it only has enough force to push soft creams through a small nozzle. In any case, these devices are very messy. If you can get them to work, the pizza itself will look pretty, but filling, using and taking apart the device creates quite the mess.
A similar argument can be made against the second type of extruder in the kitchen, the meat grinder. It will produce strands of cheese, but they will most likely stick together immediately. And even if you manage to get a standard-sized meat grinder to work with a single log of goat cheese, half of your cheese will end up as waste, and taking apart and cleaning the grinder could take longer than baking and eating the pizza.
At this softness, you might just about get away with the third option for extrusion - a spätzle press. If you get the cast aluminum cylinder type that gelaterias also use for spaghetti ice cream, and process small pieces at a time, you will end up with strands similar to the spaghetti ice cream. Again, you won't be able to separate the mass well when it falls onto the pizza, and the cleaning involves some time with a brush and a toothpick.
So, for all possible extrusion devices, you can expect that in all factors - cheese piece "niceness", cheese waste, and time spent cleaning - they are much worse than your current method of using a knife.