Antibacterial Toothpaste Recipies

On another thread, I suggested the possibility of using pine resin in toothpaste by infusing the resin into coconut oil or lard to add a bacteriostatic property to toothpaste. At least one comment suggested that infused pine resin might be too sticky for use in toothpaste, so I'm currently looking for other herbal alternatives to add to toothpaste in order to add an antibacterial or bacteriostatic property to homemade toothpaste without the risk of the paste being too sticky for practical use.

One such substance that has come to my attention is clove oil. I recently watched a video on Medieval dental hygiene where the host demostrates the use of ground cloves and salt to make a primitive toothpaste.

A while ago I also read that clove oil was also historically added to school paste to prevent it from molding over. If clove oil has strong enough bacteriostatic properties, I would expect there to be some historical toothpaste recipes that call for clove oil.

Unfortunately, it appears that the clove plant (Syzygium aromaticum) is a tropical plant, so I wouldn't be able to grow it in my temperate climate. I'm hoping there are other ingredients that have the bacteriostatic properties of pine resin and clove oil that I can either grow in a temperate climate or forage from the wild.

I use baking soda and sesame seed oil to brush with.
The oil is traditionally used in the dental practice of oil pulling, and has some antibacterial qualities.
Its also liquid at room temperature,  unlike coconut oil,  which I used to use, so its less likely to clogging pipes.

Could vinegar would have an antibacterial effect,without damaging tooth enamal?