Ibuprofen in dogs has a very narrow safety range – the risk of toxicity is high enough that we don’t encourage pet owners to use it. Ever. (And it’s deadly toxic to cats.)
Buffered aspirin can be used in dogs under the guidance of a veterinarian. Even at the correct dosage, in some dogs it can cause GI upset and even gastric ulceration.
Topical applications of lidocaine or neopsorin generally don’t hurt, but they also generally dont help. Most dogs will immediately lick it off.
Your veterinarian may opt to bandage the foot for a very limited amount of time, mostly to keep the area clean and to prevent the dog from licking. A wet wound with lots of blood flow (like a nail bed) is a great place to grow bacteria, and licking keeps it nice and moist and warm. But leaving it on too long will trap moisture and make a soggy, stinking (bacterial) mess. Please don’t bandage it yourself unless you are experienced – a too tight bandage can cut off blood flow and have serious consequences. If the dog doesn’t tolerate a bandage, an e-collar may be necessary.
Since the foot is just about as contaminated an area as you can get, antibiotics are warranted. It may heal fine on its own, but why risk it?