Does the Gen2 CX-5 come with a full size spare? Assuming that it does, I think that the difference in diameter between 225/65/17 and 255/65/17 is something like 5%. Over on the Lexus forums, people were running staggered wheels on their AWD cars, and the commonly known rule there was that there should never be more than a 3% difference in wheel diameter between any other wheel on the car. Anything more than this would damage the AWD system over time.
That said, I believe most spare tires are smaller in width and diameter than the full size tires that come with the car. As long as the spare is only used for the drive to the tire shop, and you don't go faster than 50mph (80kph) with the spare on, there should be no issue. Also make sure to never drive with a spare on one of the front tires. Move one of the rear tires to the front and put the spare on the rear if you have to.