Are you sure it is the nail that is causing abrasion between the toes? It doesn't seem possible for that to happen if the nail is short. I live in Florida and my b boy developed a sore between his toes. I bought baby socks, applied various salves (including tea tree oil and vicks vaporub) and taped the top of the sock to hold it in place. It didn't work, but it did keep my b busy trying to remove the offending sock. It turned out to be an environmental allergy. Check with your vet.
You could try washing your b's feet after running on the lanai (or going outside), but first you might need to get the sore healed before the foot washes will keep the foot healthy. Salves didn't work - we used an antiseptic shampoo to wash the feet and Mal-A-Ket wipes to clean the sore (both acquired from the vet). Then once the sore healed, I continue to wash his feet with soap/water after being outside. My b also has black spots on the knuckles on top of his paws. I thought it was dirt, but the vet said that it too was a sign of allergy.
As soon as we leave Florida. He clears up completely. Washing his feet often has really helped - no more sores. I bought a soft brush to "scrub" his toes (top and bottom) and he stands calmly in the kitchen sink (especially if I smear peanut butter or cream cheese on the counter top :->).