They have the right to look, they may not agree with the $$.
Not your problem, work with your insurer, they will settle the claim with you, perhaps pay the repairer directly.
Your insurer will then work the 'recovery' or subrogation with the other insuer.
You'll always have better service with your insurer.
Once repaired the bumper will look like new, no doubt the entire bumper cover will be repainted -not worth the time to blend.
Best of Luck