Is it recommendable to use the siwak upon entering the mosque, by analogy with the Prophet’s (peace be upon him) use of the siwak upon entering his home?

The correct view is that it is not recommended to use the siwak upon entering the mosque, as there is no authentic narration from the Prophet (peace be upon him) to support this. If it were legislated, he would have done it or encouraged his Ummah to do so. The Sunnah is to do what he did and to leave what he left. We should not do what the Prophet (peace be upon him) left. Some scholars have analogized entering the mosque to entering the home, but it has not been narrated that the Prophet (peace be upon him) used the siwak upon entering the mosque. However, if one intends to use the siwak for Salat immediately upon entering the mosque, there is no harm in that.