Is it permissible for a female veterinarian to take a two-day grant for her commute to work, knowing that her commute takes longer than that, and that the law grants her a meager amount that does not cover her transportation expenses, and that her manager sees no objection to it?

Scholars have unanimously agreed on the prohibition of a woman traveling without a mahram (unmarriageable kin) if there is fear of fitna (temptation or discord). If your early travel, as you mentioned, is considered travel in the custom of your country, then it is not permissible for you without the accompaniment of a mahram or husband.

A worker is not entitled to anything beyond their agreed-upon wage, based on the words of Allah Almighty: "O you who have believed, fulfill [your] contracts" (Quran 5:1), and the saying of the Prophet, peace be upon him: "Muslims are bound by their stipulations."

What you mentioned about a week's grant in exchange for two days of commuting is considered betrayal, deception, and falsehood, and it is not permissible unless the law permits such an incentive and there is no other way to implement it.