There is something seriously wrong here, either with the estimate, or with the appraised value.   No reputable insurance company would do an $8K repair on a vehicle that is only worth $11K.    I don't believe you have to accept their settlement to repair.    

I've unfortunately had two experiences this year - my daughter's car was written off for a repair that was about $6K (appraised value was about $11K).    The damage did not appear to be that severe BUT the estimate was over a trigger so the car was written off.    I had the option of having the repair done, but then the car would need to be re-titled as a salvage so I took the buyout.

More recently, my car was rear-ended.    Very, very minor damage (rear bumper cover had a tiny crack).    But the repair was about $4K all in (pretty sure the prices were maxed out because it was an insurance job).     So now my car is perfect, but if I were to sell it, there is a $4,000 claim on Carfax.     So I lose out when I sell or trade.