I haven't noticed any behaviour like this, but I have heard of the stories about how veterinarians and doctors see more patients on/after a full moon.
There have been tons of studies on the lunar effect (and it's funny that you use the term "lunatic" as the word "Luna" is Latin for moon), but I don't think there's ever been any reliable studies done that's shown any evidence that the moon has any effect on people's (or in this case, animal's) behaviour. My opinion is that it's an old wive's tale that's evolved over the years.
It's important to remember that everything we know now to be true, is something we've learned through science. Learned, meaning we didn't know it before. And the sun and moon have always been an integral part of our lives and culture from the beginning. Starting as our gods, then becoming creations of our gods, you can find references to them everywhere.
So in the process of finding answers to things we didn't know, hypothesis were made. For example, before we knew that hypertrichosis (werewolf syndrome) was a genetic mutation and that it was just a normal person with abnormal hair growth, someone who saw a person with hypertrichosis would during the middle ages would come up with a way of rationalizing it.
Werewolves were born from this, as people with hypertrichosis at the time would travel with circus shows at the outskirts of the cities. Meaning anyone in the city who saw them, would be at the show, or at the edge of town in the evenings/night. Not knowing any better, people rationalized it as someone who only turned into a wolfman at night, and what's different between night and day? Tada! The moon.
Nowadays we might know more about a lot of things, but psychology is a recent science that is still very young. Evolved from philosophy in the 1700-1800s people are only just learning about human behaviour, and how the mind works.
This is where some of the myths start to come into play. A popular one was that the full moon causes seizures. Part of this stems from the age-old belief in vampires, and some of it from our lack of knowledge of psychology at the time.
The thought was that since we know the moon controls the tides, and we know the human body is comprised of water, then the moon must be able to affect both things right? This thought has evolved to explain many other things. Why are there suddenly so many people making trips to the emergency on a night with a fool moon? Must be the water in the bodies being affected by the moon's gravity. It makes sense as long as you don't think about it too much.
But let's think about it. How do the tides work? It's true, that the moon's gravity pulls on the water in the ocean, causing it to rise while it's overhead, then the water recedes as the moon orbits around to the other side of the earth.
But can the moon's gravity affect the water in our bodies? It's safe to say that it doesn't. If the moon could affect the water in something as small in comparison, then in theory, we should be able to see changes in smaller bodies of water as well.
Now, to be fair, we have recorded changes in rivers, lakes, and streams, based on the moon's gravity affecting them similar to the ocean. But, the measurements are minuscule. Even the Great Lakes in the U.S. (the largest group of lakes in the world) are effected more by the wind than by the moon. They're considered non-tidal as their "tides" are around 2 inches in height. (Source)
It's important to note that everything exerts the same gravitational pull like the moon does, it's just that nothing is large enough for us to really feel it's effects over the pull that's keeping us on the ground. But, according to this article by LiveScience, the gravitational pull a child feels while being held by it's mother, is 12 million times more than what it feels from the moon - Simply by proximity.
Finally, considering tides are affected two times every day, by that standpoint it should be considered that if the water in our bodies were affected by the moon, then it should be affected twice every day with the tides, not only on a full moon.
Now, there was a study done by Colorado State university that did show that veterinarians did receive more patients around a full moon (a similar study with doctors showed no difference), but it's important to note that correlation does not equal causation. In other words, just because there are more visits to a veterinarian around a full moon doesn't mean that it's because of the full moon. An alternative hypothesis I believe is more likely, is that people are morel likely to be out at night during a full moon, because of the extra light it provides. More people, means higher chances of accidents.