The charge (raised in an xword blog on Sunday) is that "[something] may have to do with me not being too fond of forced puns..." is grammatically wrong and should read "..with MY [instead of ME] not being fond of.."
First a general principle: If you claim that something is grammatically wrong, state the rule that is being violated, not just what you think the correct version should be like--in the present case, WHY should ME be replaced by MY?
I can think of no reason b/c both versions appear correct to me. In "with me being not too fond of..." "being" is the present participle of "to be", which modifies "me", the object case of the pronoun "I", which has to be selected after "with" b/c "with" is a preposition and requires the object case (so much for people who think "between you and I" is correct--nonsense! "between you and me" is correct--but I digress).
But "with my being not too fond of.." is also correct b/c now "being" is the object of the preposition, specifically, it is a verb turned into a noun, i.e. it is a gerund, which can take a possesive pronoun like "my" as well as direct and indirect objects.
When I write comments for a blog, I prefer colloquial English (within the limitations that come with me [sic!] not being a native speaker), which by and large makes me avoid gerunds b/c they sound, to me at least, always somewhat stilted.
Google Translate is even better now
9 hours ago