your dog had an ear hematoma. DO NOT try to remove it yourself !
And some Humane Society branches have vet clinics for the public on site, which are staffed by very competent vets, and are often lower cost.
Jaybay is right; you can't remove this on your own. If a vet won't remove the drain, there must be a reason beyond what you are saying.
Do NOT attempt to remove this drain by yourself. I can understand another vet wanting the surgical report, but it shouldn't stop them from treating your dog. That doesn't sound right to me at all. If you're having problems finding a competent vet, you could try starting with your local SPCA or Humane Society. Explain the situation to them and I'm sure they'll come up with a vet who will happily get this thing safely out of your dog's ear.