Is paying a sum of money to obtain a driver's license for a mother who is unable to drive, in order to lift a ban on a customs-exempt car and acquire license plates for it, considered a bribe? And what is the correct course of action in this situation?

Respecting traffic laws is a religious obligation because they were enacted to protect lives and property. However, what necessity dictates is an exception. So, if the car will only be used fraudulently to obtain a license for your mother, there is no harm in that, as long as no damage results from it. It is permissible to use stratagems or indirect allusions (ma'areedh). If the objective is not achieved by that, then lying and paying a bribe to protect wealth are permissible, although it is better to avoid that. Every praiseworthy objective that cannot be achieved except through lying, then lying for it is permissible. And if it is obligatory, then lying becomes obligatory. The more cautious approach is to use 'tawriyah' (ambiguous speech).