I saw the Prophet ﷺ use the tooth-stick while fasting on countless occasions.
حدثنا محمد بن بشار حدثنا عبد الرحمن بن مهدي حدثنا سفيان عن عاصم بن عبيد الله
رأيت النبي ﷺ ما لا أحصي يتسوك وهو صائم
Its chain is daif. Abu Dawud narrated it in the Book of Fasting, chapter on the tooth-stick for the fasting person, hadith 2364, from the hadith of Sufyan Thawri, through the same chain. Asim bin Ubaydullah is daif due to weakness in his memory, and Ibn Khuzayma criticized this narration, hadith 2007. There is also a report on this subject from Ayesha [Ibn Majah, hadith 1677] and Bayhaqi, 4/272. Ibn Umar used to see no harm in using the tooth-stick while fasting.