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A couple of thoughts:
First is that tires deflect and the revolutions per mile is NOT what one calculates using the outside (freestanding) diameter. It's about 3% lower than that. Be sure you remember that when you do your calculations.
And, Yes! your car does NOT automatically compensate for the changing tire diameter. As a general rule, the speedometer is driven off the output shaft of the transmission. So when you change tires, you need to be aware that your speedometer and odometer are in error.
Having different tires front to rear MAY cause some emergency handling issues. The problem is that you will NOT know that until you are in such a situation - and then, it is too late. Frankly, I would make a complete changeover, rather than piecemeal.
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