Real Monk vs Fake Monk

This answer is not about the ["my son"-or-not] part, but about a monk which developed later as a "fake monk" (your term, I think it is not the best here for my example).

Would the story about Devadatta(wiki,accesstoinsight.org, palikanon.com) fit your question? Devadatta was a long time much respected bhikkhu, part of the sangha, but then built community with the son (Ajatasattu (?)) of the local king Bimbisara (the latter a friend of the Buddha) to overtake the order resp. the kingdom (the latter was successful for Ajatasattu).
One sutra is for instance A 5.100 in German in english, but there are several ones which -in my view- together show how the Buddha handled that problem.

First, Devadatta (also cousin of the Buddha) was much respected in the sangha. When now the Buddha was reported that Devadatta wanted to overtake the order, he advised his Bhikkhus to spread the word to the country, that Devadatta would no more speak for the order, that he's no more to be seen as a follower of the Buddha. Some monks were afraid (maybe even Sariputta) to tell now in the open, that the so long high valued monk would now be disrespected, because: what light would this shed on the order?! But the Buddha said: no way, we always must say the truth about the case.
Next (or before, I don't know the actual timeline at the moment), Devadatta collected a non neglectable number of followers in the sangha and made separate teachings. The Buddha seemed not to react. But Sariputta and Mogallana went to the place - seemingly let it happen, that Devadatta and his friends assumed they came as interested ones and possibly followers. Having such eminent arhats at his place they were allowed to teach, and the legend tells, that Devadatta was tired and fell asleep. While he was sleeping, Sariputta and Mogallana managed to change the mind of the monks with their talk and to lead them back to the Buddha's sangha.

Part of Devadatta's initiative had been to claim that the order's rules (for collecting food, for eating etc) were too weak, too little ascetic (today we would say "too little radical") and challenged the Buddha openly. But the Buddha defended the rules as the appropriate middle path - but left it to the taste of each one to behave individually the more ascetic way. (Ironically, that ascetic way was not what Devadatta in company with Ajatattu lived later on, if I recall the tellings in the sutras correctly).
Devadatta undertook also a life-threat to the Buddha, trying to kill him with a rock falling down the hill. The Buddha stayed "easy" after that, saying, that Devadatta would never be able to kill a Buddha. Well, this was an attack at the person, not one focusing the order, so this might be a little less relevant for your question.

In summary, what I've got from it, was that the Buddha let him do things until it would come out from the deeds themselves what's going on. Which does not mean he did him allow to proceed in secrecy - as far as the order was concerned the Buddha was not silent about the dissens, even although he had to correct a widely spread opinion about that monk and his high reputation.
And later, after Ajatattu had overtaken the kingdom (from the Buddha's friend and his own father Bimbisara) and even had put Bimbisara in jail to let him die there by starving, the Buddha still accepted (again later) an invitation of the ruthless new king - but even managed to make him regret his deeds.

The question is:

"Did he give any teachings on how to deal with monastics who abuse their position for unwholesome motivations i.e. monks amassing wealth, gaining political influence, acting as antagonists for violence."

And the answer is yes! The five books of the Vinaya aim to do exactly this.