Car crash with cop: Sue or not?

Generally speaking, the answer is wait to file suit. Filing suit within a few days after you made a claim against the City is not going to get your claim settled any quicker. In fact, it may even delay payment longer because the attorneys now get more involved. Also, all you will really accomplish is paying 1/3 or more of your insurance payment to an attorney to accomplish the same result.
It is pretty basically accepted that If you are in an auto collision, you should try to settle your auto property damage without a lawyer (even if you need a lawyer for the personal injury damages, which I am assuming are none in your case). The amount of money you will receive to repair or replace your car will be close to the same, regardless of your filing suit or retaining a lawyer.
Because your car is totaled, unless your insurance provides rental car reimbursement, you are not entitled to a rent car. Most adjusters, however, will pay for a rental car while they are deciding whether your car is totaled or can be repaired.
If the City continues to drag their feet or you need a car now, then the best strategy is to have your insurance company pay the property damages under the no fault "collision" coverage of your policy. Let your insurance company then negotiate the property damage with the City and/or the body shop for you.
Whatever you do, make sure you do not release any personal injury claims just in case you actually were injured and it simply has not manifested itself as of yet.

NOTE: THE FOREGOING DOES NOT CONSTITUTE LEGAL ADVISE OR THE CREATION OF AN ATTORNEY-CLIENT RELATIONSHIP BUT IS FOR PURPOSES OF LEGAL DISCUSSION ONLY.