I'd turn it over to your insurance. You pay them not only to pay for damage, but to negotiate how much the repair costs.
Of course you will have to pay whatever your deductible is.
i've never had an insurance company charge me a deductible when the damage is entirely on another person's car due to my fault. my deductible applies when i file a claim on damage to my own car/property (and if it's due to someone else being at fault i'm refunded my deductible when they settle the claim with the responsible party or that party's insurance company).