Actually, "Котэ" is a Georgian name. The full version is Konstantine Константинэ (კონსტანტინე), and the short informal version is Kote Котэ (კოტე). And yes, "э" is pronounceable. There was a famous sports commentator Котэ Махарадзе (Kote Makharadze - Wikipedia, the free encyclopedia).

I believe, that is the reason why some people started writing "котэ" instead of "кот" to make it funny, as if it were a Georgian person's name.