Is paying money to a police officer in exchange for the return of a driver's license and avoiding legal procedures considered a forbidden bribe, and is repentance accepted for such an act?

Bribery is forbidden and is considered a major sin. It is permissible for the briber only if it is the sole means to obtain a right or invalidate a falsehood, but it is forbidden for the recipient of the bribe under all circumstances. Based on the questioner exceeding the speed limit and the officer applying the law, he deserved what was done to him, and he is not allowed to bribe to get out of it. He must repent sincerely by abandoning the act, feeling remorse, and resolving not to return to it, for the one who repents from a sin is like one who has no sin.