I actually now have proof that Philodemus was influenced by the Stoics:
JSTOR article: Philodemus on Emotions
https://www.jstor.org/stable/43767829
I wasn't able to access the article but the Google entry says this:
QuotePhilodemus' concept of 'bites' may have been influenced by earlier Stoic presents many affinities with them. Like the Stoics, he considers 'bites' natur.