Here, @DL6ER has described a method to move the pihole logs to another location, using symbolic links. I'm sure this method can be modified to "relocate" the unbound logs to your USB stick
In this topic, a configuration is described to modify the unbound verbosity, without restaring unbound (sudo /usr/sbin/unbound-control verbosity x). You could set the loglevel to 0, and increase it at runtime, using the above command, when needed.