Is it permissible to take an amount exceeding the allocated budget, given that my manager and colleague have authorized me to do so, and is this money lawful for me?

If you suspect a shortage in your personal funds and used them in the company's project, then found an increase in the trust funds after reviewing with your colleagues, it is not permissible for you to take this increase merely based on suspicion. The default is that this increase belongs to the company unless there is clear evidence to the contrary.