Got in auto collision, do I have to tell the body shop that insurance is paying for repair?

Legal yes. But as a practical consideration it's a bad thing to do if there's a possibility that there is hidden damage. I can tell you having worked in a shop that all body shops know people do this. It's a common practice. It doesn't cause any problems unless the shop opens it up and there a whole bunch of other damage. Then you whip-out the insurance estimate and suddenly the shop has a paperwork mess on their hands. Since the damage is being paid for by the other person's insurer only your name is on the check. You can fix or not fix as much as you want. It's your money. The best thing to do when there is the possibility of hidden damage is simply to hand the estimate to the shop and ask them if they can not-fix a few things or cut corners and save you a few bucks. That way they know what the insurer initially paid for and what they didn't. They can also explain to you how they would go about doing the repair to save you a few bucks. It's much easier for the shop to start with the insurer's estimate and then rebate you for anything they don't fix. The shop can bill the insurer for additional damages that were not on the original estimate but if they don't have the original estimate from the beginning this becomes problematic.

I regret, that I can help nothing. I hope, you will find the correct decision. May be you need to handle the case without insurance, check how much your rates will increase after the claim: 1nsurance.1sthost.org

I have been in the Collision Repair industry for 35 + years and the keep in mind when working with any body shop time is money, do not waste time beating around the bush. First of all an estimate is just that, not an exactimate and chances are very good there will be additional damage found once the vehicle is torn down. Tell the shop your working with up front what your intentions are and show them the estimate the insurance company is paying off of and they should be able to give you an educated guess as to the likely hood of hidden damage. And remember, "The bitterness of poor quality remains long after the sweetness of a low price is forgotten"