I have one friend who was involved in an car accident. He was in responsible. The repair cost for the other party's vehicle costed about the same amount as yours ($3000).
The insurance company increased his premium when his insurance expeired. In 3 consecutive years the premium would increase 800 a year. No discpount for this additional $800 premium. The base premium could still be elligible for further discount.
Since you're only covered with 3rd party liability (i.e., to pay the orther party's damages), your car itself is not covered. Your loss will be further than that.
If you go through insurance company you have to pay any of the deductables and the increased premium in the coming years. But your cash flow may not be as bad as paying the entire repairing cost in all.
If I were you I would go with the insurance company if the damage is over $2500. If it were less than $1500 I would select paying myself. Any amount between 1500-2500 I would have to flip a coin.
Hope this helps.