Is it permissible to cover the treasury deficit from the monetary surplus within it, especially if a year has passed since the surplus occurred and it has been reviewed without knowing its cause?

You are a trustee, not a guarantor. If you did not neglect the safekeeping of the money and did not take anything from it for yourself, then you cannot be held liable, according to the saying of the Prophet, peace be upon him: "There is no guarantee upon a trustee." If you can compensate for what was lost with the surplus, there is no harm, unless the surplus belongs to another person or entity, in order to remove injustice from yourself. However, if you have been negligent or treacherous, then you must return what you took and replace what you neglected. The surplus cannot be counted unless you know that it is the missing amount or the remainder from previous accounts.