We have run into the same problem. And have received the same recommendation from the Vet.
We tied to slowly increase him step by step over many months. Touch feet, clip one nail, reward with high value treats. He is two now, and had his nails clipped every 1-2 weeks for his entire life. As he got older, he has become more and more against it. We never clipped too far, causing him to bleed. We tried the dremel, we tried clippers, we tried just sanding them down.
Eventually we took him to a groomer. He did alright the first few times, but again became worse and worse. The last time at the groomer, he was muzzled, foaming at the mouth, and I was basically on top of him to hold him down.
We then decided the next time to take him to the Vet. They managed, but strongly suggested giving him a med before clipping.
It has been two months since we last clipped his nails. They are getting very long. I don't know what the right answer is. I don't like giving drugs without need, but at this point we have tried the soft, slow methods, and it only became worse.
We are at the point where we fear getting bitten if we attempt it on our own.
If you find a solution, or if you try the meds let us all know. It could be some help for those of us that are going through the same thing.