Zakat is a due, sadaqa is optional, nayaj (intention) is crucial
Zakat is a due on every Muslim if his wealth reaches the nisab (the minimum amount for zakat being a due) and was one lunar year (hawl) in his possession. It has a computable fix amount and paying less is a sin while one may pay more as sadaqah, but not with the intention of zakat for the additional amount. Similarly it is essential to not confound the intention here if you intent to pay sadqah you didn't pay zakat and vice versa.
Who is eligible to recieve zakat?
Allah the almighty didn't tell us much about zakat itself in the qur'an, but He clearly mentioned the categories of people who are eligible to receive it:
Zakah expenditures are only for
(1)the poor and for
(2)the needy and
(3) for those employed to collect [zakah] and
(4) for bringing hearts together [for Islam] and
(5)for freeing captives [or slaves] and
(6) for those in debt and
(7) for the cause of Allah and
(8)for the [stranded] traveler - an obligation [imposed] by Allah .
And Allah is Knowing and Wise.
Anybody spending zakat for someone or something out of these 8 categories is doing a sin. Because he disobeys Allah.
Further scholars even consider it preferable to spend it in this order, so if you know poor people, you should spend it to them before spending it to a person fighting for the cause of Allah!
Zakat shouldn't be given to mosques and alike
Therefore scholars are in total agreement that spending zakat for a mosque, for its re-/construction etc. is not permissible. The same basically applies for other institutions like hospitals -as mentioned in fatwa islamweb #20886 (in Arabic)- and any kinds of organisations (except for the case of those who deliver zakat to the people -and even here one should be careful as it is always best to give your zakat yourself).
On the other hand it is perfectly permissible to spend it to people who fight for the cause of Allah. Some people misused this as an argument to explain that your request might be permissible, but doing something for a mosque is far away from fighting for the cause of Allah even if this mosque is al-Masjid al-Aqsa. This argument and its refutation was mentioned here in fatwa islamweb #5757 -in Arabic-, but they added that it could be considered in a case of dire necessity where no other option seems applicable.
What is possible
Further you may know that beside the obligatory zakat you are allowed to spend money as sadaqah etc. and further instead of spending it for the mosque or its committee or an organisation which supports it it is much better to spend zakat and sadaqah for the needy people and those fighting for the cause of Allah in its surroundings.