Going through the other party when they're a commercial entity can be a lot of headaches. Your own insurer's only requirement is that the damages are the "result of a collision", which they are. Your own insurer will just pay for it and pursuing the other party becomes your insurer's problem and not yours.

Your personal auto policy doesn't have a liability deductible but some commercial policies do, so if they have one, and it's large enough (like they have to eat the first $5k or $10k of your claim), you'll possibly end up dealing with somebody who is just trying to hand you some cash so they don't have to push another claim through their insurance.

It looks like you need a bumper cover, side molding, and a couple tail lights. The quarter panel is 50/50 but if it was mine I'd ask the shop to try to fix it before going with a replacement. A good metal guy should be able to do it and keep them from having to respray the tailgate and right rear door for color match.