While this will not help now though this might help future dog owners or folks thinking about getting a puppy.
When the puppies are young play with their feet by rubbing or stroking. Getting them used to the feet being handled.
I did the same thing with the kitten when we got her.
Nowadays anytime I pick her up I stroke her feet.
Treats will help distract the dog.
This article talks about using lavender oil to calm the dog:
Aromatherapy relaxes and helps your dog fall asleep. Use some lavender oil for massaging your dog to calm it down. Make sure to rub the back of its head using your hands.
Don’t forget to stroke down the base of the spine. Massaging helps stop the production of cortisol, reducing stress on your dog.
When your dog is calm enough, you can prepare your clippers, nail file, or nail grinder.
https://dogsbestlife.com/dog-grooming/nail-clippers/