All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is permissible for a fasting person to brush his teeth during the day in Ramadan with the known siwak or anything that cleans and purifies the mouth and teeth.
That is because Imam Al-Bukhari reported from ‘Aamir ibn Rabee‘ah, who reported from his father that he said: “I have seen the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, more often than I could count, using a siwak while he was fasting.” [Ahmad and Abu Daawood]
As for the matter of bleeding while using the siwak, it has no effect on fasting at all.
Allaah Knows best.