Ok yes there are some pieces from Levon I that have the cross with a crescent in each quarter (e.g. https://en.numista.com/catalogue/pieces115204.html , which is somewhat close to your piece). I don't have any good references for Cilician Armenia, but hopefully someone else here does.
"If you climb a good tree, you get a push." -----Ghanaian proverb
"The danger we all now face is distinguishing between what is authentic and what is performed." -----King Adz