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 man to pay his ex-wife her due dowry from zakat money if he has no money other than his income from which he provides for his family, as he is considered as one in debt. The indebted are among the rightful recipients of zakat in Islam. Allaah, the Exalted, says (what means): {Zakat expenditures are only for the poor and for the needy and for those employed to collect [zakat] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allaah and for the [stranded] traveler – an obligation [imposed] by Allaah. And Allaah Is Knowing and Wise.} [Quran 9:60]
The category of the indebted includes whoever borrows money to perform a permissible act and has no money to repay his debts. Indeed, the basic principle is that borrowing money to pay dowry is permissible in Islam.
Allaah Knows best.