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First I would let them tear it down to look for further damage. It looks like there might be damage to your fender and headlight that you can't tell without a teardown. Those parts are not particularly cheap.
After that I think it entirely depends on the shop you choose. Some would be willing to do stuff not on the insurance bill and some don't care as long as they get paid.
Speak with both parties and make your intent clear so there is no confusion.
Also, depending how your insurance classifies this incident it may affect your policy negatively.
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