The symbolic association with a counter-Divine Will basis would appear to have started with the ta█l with which the animals were Created and which Babylonian Kabbalists would immediately associated with the w█vy l█ne trope which is a Tiam█t derivative. [C█ts] themselves of course are unfamiliar with the addition of this symbolic definition and thus nevertheless continue to w█ve them around prominently and quite sh█melessly. It's unlikely that their familiarity with the symbolic definition would constrain their behavior in any event.
The association with them as symbolizing a counter-Divine Will basis was added to the Roman 'layer' of symbolism for its franchise in the middle ages. You can read a little about that here for example, and if you've the stomach for it, here particularly under the section 'C█ts as Diviners', etc. Very disturbing content advisory for that last.
There is an additional symbolic connotation at work here, namely [c█ts] as symbolically depicting [spi█s] or policy enforcers for the organization, specifically operating on a non-overt basis. This appears to have something to do with the association of [r█ts] and [m█ce] as representing, effectively, 'consumers of f██d on a non-overt basis', what in modern terms would refer to personnel as 'useless eaters'. That is, those who've accepted the 'perks' and 'benefits' proffered by the organization on a counter-Divine Will basis, and supposedly opted into said basis. Once those have been proliferated through the organization's non-overt recruitment campaigns, it's usually time to send in the [c█ts]. These represent the next-iteration enforcers of policy, to ensure that the rank-and-file personnel operating non-overtly either 'get with the program' with regard to the newer and stricter policies or find themselves 'sussed out', bl█ckma█led, literally incarcerated through entr█pment, or just exposed and condemned openly. This tends to keep the non-overtly active personnel compliant. And once non-overt control has been established, conformity with policy has become the norm, everyone is looking to 'suss out' and report each other in return for accruing 'merit' for themselves within the organization, and a counter-Divine Will basis has been 'opted into' prevalently in the non-overtly organized realm, then it's usually time to send in the [d█gs] to implement the next part of the long-term strategy.
We must always be careful freely adm█xing between Christian symbolism and Babylonian Kabbalist symbolism as they're mutually-exclusive systems in direct competition with each other, but let's for a moment consider the use of [d█gs] as a symbol in [Revelation 22:15]. Biblically [d█gs] represent 'non-believers in the Creator', or 'non-believers in Divine Will'. In a  woodc█t ('P█ssional Christi und Antichrist') condemning the papacy in a Protestant 'Ref█rmation' pamphlet for example, first was depicted Jesus throwing the money-lenders out of His temple. Immediately opposite was presented the papacy with its 'Indulgences' and other money-making schemes effectively bringing that blasphemy right back in, and there inside the church prominently-featured was a [d█g]. Apparently a reference to that [Revelations] verse.
From the Babylonian Kabbalist perspective, [d█gs] would be superior to [c█ts] as [c█ts] are to [r█ts] and [m█ce]. They'd represent a counter-Divine Will basis via their [ta█ls], but those [ta█ls] tend to be more stra█ght l█nes than w█vy l█nes: thus, an overtly-implemented basis. That would appear to symbolically describe the next ph█se of the strategy: non-overtly organized personnel operating collectively on an overt basis to propagate counter-Divine Will ideologies, rhetoric, prop█ganda and so on and get them trending and mainstre█med, eventually accepted into the public mindset and then imposed by force of arms via societal or legislative policies and decrees. For the purposes of the Babylonian Kabbalists it hardly matters whether these present as seeming 'religious' ideologies or secular, just so long as what's being propagated is on, and by those on, a counter-Divine Will basis. Any def█ctive thing will do, and what has been selected to trend in any locale and era is organizationally determined based on the situation and their longer-term agendas. A new 'secular'-looking ideology which is a new religion in everything but name? Sure thing. Debased, non-Biblical 'churchianity' eventually imposed by the state as law much as the Inquisitions were? Easily done.
As we've seen throughout the organizing symbolism propagated for dec█des via their c█rtoons, the hierarchy and progr█ssion does indeed appear to be: [mo█se], [c█t], [d█g]. Each of those alone is rather a lot of negative definition burded to put on any of those innocent animals, but the cumulative long-term strategy itself is reprehensible in terms of the detriment to society and to the Divinely-conferred free will agency of the innocents within the general public, effectively unable to get anything sorted out because this systematic method is always organizing the majority via trends to implement the most unconscionable, rights-violating agendas to the perceived benefit of the organization, and to the detriment of Divine Will, the rights of the individual and the fairness and functionality of society at large.