Peace be upon you. It is not considered a breaking of the fast to clean your teeth with a toothbrush and toothpaste, as long as the fasting person does not swallow anything that mixes with their saliva during the cleaning process. A small amount that is consumed with saliva does not harm the fast.
According to Grand Ayatollah Sayyid Sistani (may Allah protect him): It is permissible to use a toothbrush and toothpaste as long as nothing is swallowed, even if one feels the taste in their mouth.
Source: [Official Website](
https://www.sistani.org/arabic/qa/23878/)
According to Grand Ayatollah Sayyid Khamenei (may Allah protect him): It is permissible to use a toothbrush, toothpaste, and miswak to clean the teeth, provided that nothing is swallowed, even if one feels the taste in their mouth.
Source: [Fiqh Al-Wali: Rulings of Ramadan: 10, 11](
https://www.baqiatollah.net/article.php?id=3549)
May you be successful in all good.