A more mainstream English word for "invoking Poe's Law"

Let me start by saying that I don't think invoking Poe's Law quite fits in your blank. You're not invoking the law, but you may be demonstrating or exhibiting or corroborating it.

The closest phrases I can think of that fit your situation are blending fact with fiction and can't tell what's real.

For example:

I swear I was only doing it ironically, but people kept telling me I was blending fact with fiction.

Unfortunately, blending fact with fiction is sometimes used to describe the process of mixing falsities into the telling of a true story. But understood in a different way, it clearly applies to your hypothetical person who does sarcastic and ironic bits so often and so seemlessly that others cannot tell what (of his) is sincere.

Or:

I swear I was only doing it ironically, but people kept telling me they couldn't tell what was real.

The phrase can't tell what's real is rather common, and it perfectly describes your hypothetical audience.

"I swear I was only doing it ironically, but people kept telling me I was becoming it."

Become — ODO

verb 1.1 Grow to be; develop into

Or, turn out to be

Related quotes:

"Be careful of your thoughts, for your thoughts become your words. Be careful of your words, for your words become your actions. Be careful of your actions, for your actions become your habits. Be careful of your habits, for your habits become your character. Be careful of your character, for your character becomes your destiny.

"The mind is everything. What you think about, you become." - Buddha
"You become what you think about all day long." ― Ralph Waldo Emerson
“Beware that, when fighting monsters, you yourself do not become a monster... for when you gaze long into the abyss. The abyss gazes also into you.” - Friedrich Nietzsche

Janus-faced

2. having two contrasting aspects, as the alternation of mood in a capricious person.

3. two-faced; deceitful.

5. having or containing contrasting characteristics

Two-faced

2. deceitful or hypocritical.

Double-dealing

2. using duplicity; treacherous.

Duplicitous

1. marked or characterized by duplicity.

Equivocal

1. allowing the possibility of several different meanings, as a word or phrase, especially with intent to deceive or misguide; susceptible of double interpretation; deliberately ambiguous: an equivocal answer.

2. of doubtful nature or character; questionable; dubious; suspicious: aliens of equivocal loyalty.

3. of uncertain significance; not determined

Ambiguous

1. open to or having several possible meanings or interpretations; equivocal: an ambiguous answer.

Indiscernible

not discernible; that cannot be seen or perceived clearly; imperceptible.

Double-edged

2. capable of acting two ways or having opposite effects

[All definitions from "Dictionary.com Unabridged Based on the Random House Dictionary"]

"I swear I was only doing it ironically, but people kept telling me I was playing it a little too close to the cuff."

close to the cuff

It means that a person keeps their motivations a secret. It comes from poker: you hold your cards close to your "vest" or body so you don't "tip your hand" or accidentally allow the other players to see your cards. –ELU

In this instance, they've cried wolf too many times and we're no longer sure if they're faking it. Because they like to play this game a little too close to the cuff.

Keeping with the spirit of Poe's Law and to use a single word, I'd reword to include facetious.

"I was only doing it ironically, but people kept telling me that they couldn't tell if I was being facetious."

facetious /fəˈsēSHəs/ adjective

treating serious issues with deliberately inappropriate humor; flippant. –Google