Is $200 a reasonable settlement for whiplash?

The settlement offer of $200 is ridiculous and should be rejected.  That won't even cover your initial visit to the chiropractor.
There should not be any settlement of your case until you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary, which means having reached a point in your medical treatment where no further improvement is anticipated.  Since your medical treatment is ongoing, you don't know your total medical bills yet and need that total along with your final medical report in order to discuss settlement.  You will also need your total wage loss after completing your medical treatment
When you complete your medical treatment and are released by the doctor or are declared by the doctor to be permanent and stationary, obtain your medical bills, medical reports and documentation of wage loss.  Your personal injury claim filed with the at-fault party's insurance carrier should include those items.
Compensation for the medical bills is straight reimbursement. The medical reports will document the nature and extent of your injury and will be used to determine compensation for pain and suffering which is an amount in addition to the medical reports.  Compensation for wage loss is straight reimbursement.
If you complete your medical treatment and fully recover (no residual complaints), I would ask for quadruple the medical bills to compensate for pain and suffering, but NOT expecting to get that.  That would be a starting point in settlement negotiations.  The insurance company will respond with a much lower offer and you can continue negotiating.
If you are released by the doctor and have residual complaints, depending on what those complaints are, I would ask for much more than quadruple the medical bills to compensate for pain and suffering, but again expecting a low offer in an initial response from the insurance company.  If the medical report says that you will have future complaints, the estimated cost of future treatment discounted to present value should also be a factor in the amount you are receiving as a settlement.  Once the case is settled, you can't go back to the insurance company and ask for more money in the future.  When the case is settled, you will be signing a release of all claims in order to receive the settlement check.
If you are dissatisfied with settlement offers from the at-fault party's insurance company, reject the settlement offers and file a lawsuit for negligence against the at-fault party.
If the case is settled, NO lawsuit is filed.
If the case is NOT settled, your lawsuit for negligence against the at-fault party must be filed prior to the expiration of the applicable statute of limitations or you will lose your rights forever in the matter.
Since the accident happened only a week ago, any talk of settlement is premature.  You may need months of medical treatment (physical therapy) before you are released by the doctor upon completion of your treatment or to reach a point where you are permanent and stationary.