Like petty thieves that pickpocket in any kind of convoluted situation, like a street fight and prey on innocent onlookers.

It looks those fat cats hold specific positions in society, what makes me think more in terms of peces gordos (figuratively, big predatory fishes at the top of the food chain) or maybe it is a fat cat el que corta el bacalao (those with power to allow -or not- poor people to get their daily bread: el/la mandamás - los mandamases).

That idiom's pescadores are not specifically spotted or alluded there otherwise than as generic third parties who benefit from internal disturbance. They perfectly may be opportunistic petty criminals, like those mentioned by chileno:

or even those who exploit that divide et impera on an industrial scale, but it also includes circumstantial beneficiaries that are law abiding and morally oriented.

Then, there is no matching idiom in English. I think a term that can be used is inadvertant beneficiary.

Oh no, those we call sharks, or even politics.