A) You suggest that the contractor has a contract with the insured. Normally money has to exchange hands before there is a valid contract.
B) If the contractor has a subject-to agreement or any other type of non binding agreement, even if signed by the insured, it is , (in most states) not valid.
C) Your scope of loss should be presented to the company with a notation that the insured nor their potential contractor do not agree. Suggest to the carrier that they send a reservation of rights letter to the insured, that will make the insured aware of the seriousness of the situation.
D) The contractor's estimate is just that , "an estimate", just as is your scope of loss. Many times items are under or over priced on a unit price basis, and that may be a reason for a supplement, if this would occur.
Sounds like you have an uncooperative insured, being led by the nose by their contractor. As stated above your assignment is to close the file.
You do this by preparing a proper scope, submitting it for payment and explaining in detail what has transpired. You do NOT have the authority to settle the claim, only the carrier can do that.
If you follow the rule of law and address the 3 "C's", Cause, coverage and cost you have done your job.