Using toothpaste whilst fasting
In the name of Allah, the most Beneficent, the most Merciful.
The use of toothpaste whilst fasting is Makruh Tahrimi. There is an increase chance of a person swallowing the paste, which will result in the fast breaking. Try to use other alternatives such as Miswak or a toothbrush (without the paste) and brush at regular intervals during the day to stop the bleeding and pain.
However, if the aforementioned alternatives come to no avail then Mufti Khalid Saifullah Rahmani (DB) has said that it will be permissible to use toothpaste if there is a need (uzr). Care should be taken that you do not swallow the paste. (Jadeed Fiqhi Masail p.102 v.1)
Finally, I would like to stress this is a specific ruling for this particular situation and is not applicable for every individual.
Only Allah Knows Best
Mohammed Tosir Miah
Darul Ifta Birmingham