The siwak is recommended at all times, during fasting and otherwise, at the beginning and end of the day.
The evidence for that is:
1- Al-Bukhari (887) narrated on the authority of Abu Hurayrah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: (Were it not that I would cause hardship for my ummah or for the people, I would have commanded them to use the siwak with every prayer.
2- Al-Nasa’i narrated on the authority of Aisha (may Allah be pleased with her) that the Prophet (blessings and peace of Allah be upon him) said: (The siwak purifies the mouth, (It pleases the Lord) Narrated by al-Nasa’i (5). Al-Albani authenticated it in Sahih al-Nasa’i (5).
In these hadiths there is evidence of the desirability of using the siwak at all times, and the Prophet (peace and blessings of Allah be upon him) did not exempt the fasting person, rather the generality of the hadiths includes the fasting person and the non-fasting person.
It is permissible to swallow saliva after using the siwak, unless something has dissolved in the mouth from the siwak, in which case he should remove it and then swallow his saliva. It is also permissible for the fasting person to perform ablution, then spit out water from his mouth and then swallow his saliva, and he is not required to dry his mouth from the water from rinsing his mouth.
Al-Nawawi said in “Al-Majmoo’” (6/327):
Al-Mutawali and others said: If the fasting person rinses his mouth, he is required to spit out the water, but he is not required to dry his mouth with a rag or the like, without any difference of opinion.
Al-Bukhari, may Allah have mercy on him, said:
Chapter on the siwak of the fasting person, whether wet or dry. . . Abu Hurairah said, on the authority of the Prophet, may God bless him and grant him peace: “Were it not that I would make it difficult for my Ummah, I would have commanded them to use the siwak before every ablution.” Al-Bukhari said: He did not specify the fasting person from others. Aisha said, on the authority of the Prophet, may God bless him and grant him peace: “The siwak purifies the mouth and pleases the Lord.” Ata' and Qatada said: He swallows his saliva. Al-Hafiz said in Al-Fath: With this translation, he indicated a response to those who disliked the fasting person using a wet siwak for cleaning the teeth... Ibn Sirin's analogy of the wet siwak to the water used for rinsing the mouth has already been mentioned. .
“And he did not specify a fasting person from others” meaning, he did not specify wet from dry. With this statement, the suitability of all that he mentioned in this chapter for the translation becomes clear. The comprehensive of all of that is his statement in the hadith of Abu Hurayrah: (I would have ordered them to use the siwak at every ablution), as it requires that it is permissible at all times and in all circumstances. .
(Ata’ and Qatada said: He swallows his saliva) Its relevance to the translation is that the most that is feared from the wet siwak is that something of it will dissolve in the mouth, and that something is like the water of rinsing the mouth, so if he spits it out of his mouth, it will not harm him after that to swallow his saliva. The words of Al-Hafiz Ibn Hajar ended briefly.
Sheikh Ibn Uthaymeen said:
The correct view is that using a miswak is a Sunnah for the fasting person at the beginning and end of the day. End quote from Fatawa Arkan Al-Islam, p. 468.
(The miswak is a Sunnah for the fasting person throughout the day, even if it is wet. If he uses a miswak while he is fasting and finds heat or something else in its taste, then swallows it or takes it out of his mouth with saliva on it, then puts it back and swallows it, it does not harm him. Fatawa Al-Saadiyyah 245.
(He should avoid anything that has a substance that dissolves, such as the green miswak, and anything to which an external taste has been added, such as lemon and mint. He should remove whatever has disintegrated from it inside the mouth. It is not permissible to swallow it intentionally. If he swallows it unintentionally, then there is nothing wrong with it.)
And Allah knows best.