The word ‘Pilish’ occurs neither in Benson, nor Wells, nor Vikivortaro, nor Vikipedio, nor ReVo, nor Sonja. Admittedly, it does not occur in Merriam-Webster nor in Wiktionary either, but I don’t see any reason why it couldn’t or shouldn’t. It does occur in Wikipedia, however. I was unaware of the existence of this word until I saw it mentioned by Dario in answer to my question about a mnemonic in Esperanto for the first 8 digits of pi. My guess is that ‘Pilish’ in Esperanto would be ‘la pila’.
"see any reason why it couldn’t or shouldn’t"
I see a reason: The lexicon of Esperanto must not be bloated to the unbelievable size English has. There is also no need for an extremely rare notion like Pilish to have some short expression.
Esperanto has the good tradition of forming words for concepts by composition, which is much clearer than mere borrowing (*pila, which additionally would give an awkward homonym) or also affixation (piuma, which still needs clarification).
For this reason I'd like to propose something like Pi-vortigo/-frazigo.
In Esperanto, la pi-uma fits in several ways. First, -um- can be used to espress any unclear and unusual relationship with the root; second, the verb umi is already used to express obsessive, pointless activities like this one; third, it consists of pre-existing elements only. It is just the type of word which would be created spontaneously when two Esperantist nerds discuss such things in a gufujo somewhere (maybe comparing la piuma angla with la piuma hungara...)
The word Pilish was created in English for fun and as a joke. Obviously, in a serious context you would use the pedantic π-frazigo (as in “encoding of π into sentences”.) Neither π-frazigo nor piumado would be understandable outside context, and one can choose the appropriate register in every context.