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I know that most Poskim forbid it. If my hands are dirty, then I am permitted to remove the dirt with water and soap if necessary, preferably up to the knuckles if possible.

Now that we know that our teeth are actually dirty with plaque.

From Wikipedia

Components of plaque

Plaque consists of microorganisms and extracellular matrix.
The microorganisms that form the biofilm are mainly Streptococcus mutans and anaerobes, with the composition varying by location in the mouth. Examples of such anaerobes include fusobacterium and actinobacteria.
The extracellular matrix contains proteins, long chain polysaccharides and lipids.
The microorganisms present in dental plaque are all naturally present in the oral cavity, and are normally harmless. However, failure to remove plaque by regular tooth brushing means that they are allowed to build up in a thick layer. Those microorganisms nearest the tooth surface convert to anaerobic respiration; it is in this state that they start to produce acids.
Acids released from dental plaque lead to demineralization of the adjacent tooth surface, and consequently to dental caries. Saliva is also unable to penetrate the build-up of plaque and thus cannot act to neutralize the acid produced by the bacteria and remineralize the tooth surface.
They also cause irritation of the gums around the teeth that could lead to gingivitis, periodontal disease and tooth loss.
Plaque build up can also become mineralized and form calculus (tartar).

I understand that on Yom Kippur we have an additional issue of Inuy, afflicting oneself.

ילמדינו רבינו
Why isn’t plaque considered like “dirt” that may be removed?
It could be argued that nobody, even a dog, would swallow tooth paste let alone listerine.
Why not allow brushing with a half a cupful of listerine or similar?