I hope you are feeling better and your car gets sorted out.
I got hit from behind, luck would have it that I saw the idiot behind me the split second before he hit me so although I didnt have time to brace myself I knew what was comming, my missus didn't have that luxury though and we had a bad necks afterwards for a couple of days.
The damage we sustained (to our mx5) was way worse than yours and we didnt claim neck injury (even though the guy who caused the accident acted like a tw*t afterwards) because, in all honesty, the stiff neck wore off after a couple of days so I wouldn't claim for an injury I didn't have.
My dad got hit from behind in a moggy minor and he actually bent the steel frame in the non reclining seats with his back to a virtually lying down position. The engine in the renault that ran into him fell out and it's driver couldn't close the doors afterwards as the whole car was bent and had to be scraped off the road, that give you an idea of the force of the crash, he had a bad back for a week but didnt claim (or complain about the pain the crazy sod) and certainly should have!!
These days it seems that it is implied that you must claim for personal injury no matter how minor. imo I'd say if you are genuinly uncomfortable then definately make a claim, but if it's just a sore neck that passes after a day or two I think it's a bit missleading to make a claim.