What is the ruling on purchasing a car from an interest-based bank out of necessity, in a non-Islamic country, given the unavailability of transportation for work and daily needs?

If the usurious bank will lend you the price of the car with usurious interest, then this is not permissible except in a case of necessity that allows committing the prohibited act. Necessity is when a person reaches a point where, if they do not partake of the forbidden, they will perish or be close to perishing. What you mentioned, that the car rental fee equals the price of a new car, and that your work requires a means of transport, none of this is considered a necessity that permits committing the prohibited act.