It's a scrumper piercing, through the frenulum (the connective tissue between the top gums and the upper lip). You can hook a curved metal rod through the piercing with spikes at the end to resemble fangs. They're pretty popular lately in Japan for alternative fashion.
It seems to be a parody of the chuuru cat food tubes, yes.
"Kitty cat" in this instance is a separate noun (rather than an attribute of the "chururu"), but it's not a complete clause so the full meaning isn't clear. It's closer to "give a cat a chururu" but that's speculative.
However chururu is also an onomatopoeia for sipping or slurping something (like drinking through a straw or slurping up noodles) and in that case, the sentence would be closer to "slurp a cat". This onomatopoeia is also where chuuru gets its name from.
That double meaning seems to be the joke. But the author's mind is an enigma so who can say what she meant.