In Michigan you may not sue the other owner or driver for the loss of use of your car or for a rental car. You must buy that optional coverage from your own insurance company to be covered. So you cannot collect from the at-fault party for the cost of your rental car as part of the mini-tort amount you go after.

You can get up to $500 using the mini-tort law however if your collision deductible is less, which it is at $250, that is all you can collect according to the Michigan insurance regulator.

You can file under the at-fault party's Property Damage Liability and there is no deductible attached to that coverage.

A mini-tort lawsuit is a remedy in which you can get up to $500 for the damages the at-fault driver caused to your vehicle. You cannot recover for the rental car expense due to the fact that is optional insurance coverage you would obtain through your own insurance policy. You mentioned that you had not purchased this type of coverage as part of your auto insurance policy so you could not make a claim for it with your own insurer.

Michigan law section 500.3135 discusses mini-tort, specifically subsection 3(e) if you would like to read more about it.

You should be able to use the mini-tort provision to sue the at fault driver for the deductible ($250) associated with your Collision coverage but you cannot recover for the cost of a rental car to use during the time your car is being fixed since you do not have that coverage through your own insurance policy and cannot sue for it under the "mini-tort" law.