That's a hard one - I would want the dangly bit of the toenail gone, but that can be quite a painful experience to remove it.
A vet trip will most likely end in the same outcome, he will be restrained and the nail will be removed without pain relief (most likely trimmed off) a temporary bandage to keep it clean may be applied but they just seem to rip these off.
Nails are amazing self-repairing things and my cat once broke all (I mean ALL) her back claws off from gripping the road so hard after an altercation with a car after she got out of the house. The vet said that being inside would keep them clean and that although it looked awful (bloody and swollen back paws, she could hardly walk) they would repair and re-grow extremely quickly - He was right. Within a couple of days she was back to acting normal and the claws grew back really fast. No pain relief was given and no antibiotics (which surprised me, this particular vet was always the money grabber) but sure enough, he was right and she bounced back quickly.
I would see if you can give it a good wash with lightly salted water and a good dry, to see if you can assess the nail better - in order to determine if the broken piece is something you can remove yourself - before your dog has a go himself.
They do heal very very quickly and I would also try to avoid a vet visit for something like this, as nail injuries are very common but very fast healing, but of course every injury is unique in whether it will require a vet visit.
Asha - 22/05/15 Wolfhound cross a Dogue De Bordeaux X American Bulldog.