It's hard to say without visually inspecting it, but from the pics, it doesn't look this bad to where you need a battery/AC recharge and possibly "misc" items. Since the car "drove" off 10-15 feet into a container, I couldn't imagine the the truck going more than 10mph by itself, which means the amount of damage you caused should be very.. very.. minimal. Maybe a dented bumper, broken bumper tabs, broken headlight tabs... but that's about it.

While you are also worried about getting into another accident later and insurance most likely not paying for the parts to be replaced if you had an accident in the same area, I would just let insurance handle it if you are very worried.

That said, I would recommend you go to a few more shops and get a few different quotes. It sounds like the first shop is just trying to make some more money from you, or there is legitimately more damage than what the pictures show.

If you are mechanically inclined:
Take the time to remove the front bumper and inspect the area yourself. If at the end of the day, all you need is a new headlight, maybe a new hood and brackets, I would just go to a junk yard or buy used parts online and repair it myself. Hell, the crack on your headlight tab could be fixed with some JB weld too. I personally would go this route mainly because it's already a 15 year old truck, and putting $5k just to put it back to stock doesn't make sense to me. If it were $5k in mods, it'd be a totally different story. However, I am quite mechanically inclined, so my answer may be a little skewed.