What are our rights and settlement entitlements in a hit-from-behind incident?

If the other driver is at fault, as this one apparently is (a driver who rear ends another is almost always at fault, since he or she has the responsibility to maintain a safe following distance and speed, and to pay attention so as to stop in time), he or she is liable (or responsible) for the injuries and damage he or she causes. If the driver or his/her insurer will not offer what you consider fair compensation, your right is to sue the driver (you sue the driver, not the insurer) for your losses. You can potentially recover:
1) The lesser of the then-current fair-market value of the car--basically, its Blue Book value at that moment, given make, model, year, condition, mileage--or the cost to repair. The law does not require someone to pay to fix something if the cost to fix is more than the thing is worth, so if the value of your car is less than the cost to fix it, the other driver would only be liable for its value, not the repair cost.
Whether or not you had a car payment before or will have one after is irrelevant: the law looks to the car's value or its repair cost, not to what you did pay for it, or would have to pay for a new car.
2) Lost wages, if your son has missed work due the injury.
3) Your son's out-of-pocket medical costs (i.e. costs not paid by insurance, etc.)
4) If your son suffers siginificant life impairment or disability, lasting at least *many* weeks or months, he may be entitled to something for "pain and suffering."
5) If you believe that your own insurer, under the terms of your policy and the coverage you bought, should have paid compensation to your son, you could sue them for breach of contract, for not honoring their policy (which is a contract) obligations. But note that they only have to provide the coverage called for in your policy; they don't need to do anything more than the policy you purchased requires.
Note that if you sue, you will need experts for part of your case. For example, if you think they are undervaluing your car, you'd need an insurance adjustor or car appraiser to testify as to value--your testimony, as a layperson, will not be good enough.
Or if your son sues for pain and suffering, he'll need a medical expert to testify about his injuries--how they caused, the effect, the prognosis, etc.