Praise be to Allah.
It is not essential to remove this brace, because rinsing the mouth means making water move around in the mouth, and this can be done when this brace is there. But if it is easy to remove it – without any hardship – when doing ghusl for janaabah, then that is better.
Shaykh Ibn Baaz (may Allah have mercy on him) was asked:
I have dentures. Is it acceptable to remove them for one wudoo’, or must I remove them for wudoo’ every time I do wudoo’?
He replied: There is no need to remove them. You can rinse your mouth when doing the first wudoo’, and so long as you remain in a state of purity, you may pray with that wudoo’. Then when you want to do wudoo’ again, you can rinse your mouth with the dentures in place, and there is no need to remove them." (Fataawa Noor ‘ala ad-Darb 5/101).
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked about someone who has some false teeth which he is able to remove, because they are not implanted, but it is difficult for him to do that. Is there anything wrong with that with regard to wudoo’, i.e., rinsing the mouth?
He does not have to remove them when rinsing his mouth, because they are small, and it is sufficient to move the water around in the mouth, and because the water will most likely get in, for water – as you know – is liquid and will easily get in between these false teeth and the gums. But if he removes it, especially when doing ghusl for janaabah, that is better." (Liqa’ al-Baab al-Maftooh 158/20).
Allah knows best.