Which is better: to perform the 'Asr prayer in the car with my father before the optional time expires, or at home during the time of necessity, knowing that returning by public transportation would be difficult?

It is not permissible to delay the Asr prayer until the time of necessity (when the sun turns yellow) without a valid excuse. Your engagement with study classes is not an excuse that permits you to delay it, and it is not permissible to pray sitting in a car except for an excuse that prevents standing, and what you mentioned is not an excuse.

You must strive to perform the prayer at its preferred time by seeking permission to pray, or by performing ablution immediately before the prayer and performing it at once in a way that does not inconvenience your father.

If that is difficult, then advise your father. If he insists on preventing you and you have no choice but to ride with him, then there is no blame upon you for praying after the time of necessity has begun; this is an excuse for you. Do not pray in the car as long as you can catch the prayer within its time, even if it is the time of necessity. However, if you fear missing the prayer entirely and your father refuses to let you perform it, then pray in the car according to your situation.