I'm sure we could debate the validity of counselors administering an MMPI, and in fact you'd probably have the edge there because your training on that test probably far exceeds my own.
However, IMHO its pretty much obvious that they should be able to administer a HAM-D.
Just about any medical personnel can administer an injection, write a drug name on an Rx pad or give advice about that nagging cough. We all know that nurses write or call in a lot of prescriptions without legal Rx authority. It's the greater issue of when to use a test, how to interpret it, what to do with the results, and how it fits into a larger picture.
That being said, the issue of who can do psychological testing is indistinct and complicated, just like psychology! Any knucklehead can "administer" an MMPI and cut-and-paste the results into a "report". The question is how we try to enforce some kind of quality in psychological services, and limiting some of the testing to those with a sufficient training is one way, but that system has more holes in it than your average sieve.
Many people who don't understand the psychology business, including some psychiatrists, think all you have to do is get the printout and it tells you if the person is depressed or a borderline. And some of the test manufacturers make good money catering to the people who just want to be told what it means without having to know too much.
I've never heard of Ed.D.'s being limited in their practice scope, or an attempt to do so by APA, although their training is very thin on psychometrics. (APA's current political model is to have as many people under their tent as possible regardless of the quality of their training or practice.) The same goes for the counseling psychology degrees. They were originally designed to counsel in vocational and educational settings, and not perform clinical assessments or deal with severely ill populations. But the law usually says they and Ed.D.'s are the same as those trained in clinical psychology.
And many of the professional schools which purport to give a clinical psychology degree also give short shrift to this kind of training. They teach the what but not the how or why, saying that's "science stuff" and they aren't training "researchers". I had one professional school student, a star in her program, who could not tell me what a t-score is. She had memorized what the cutoff was on the MMPI scales, but she had no idea what the score was, or why it was the cutoff. This is like saying "I know that an A1C of 10 is bad, but I have no idea what an A1C really is."
And to be fair, I've met psychologists from good Ph.D. programs I wouldn't send my dog to. You've met plenty of M.D.'s who are the same.
So can a counselor administer a Ham-D? Sure. And that person might use it very well, or misuse it. Does the counselor understand the cutoffs and what they really mean? Does she understand depression and how to distinguish the cognitive, affective and physiological symptoms and the various factors that may influence them? Or does she just say "this patient is depressed because his score is an XX." Does she choose the HAM-D over the Zung or the Beck or the DEP scale of the PAI or Scale 2 of the MMPI for a reason? They each have their advantages. And so forth ...
The answer is that the odds of properly using psychological testing increase with the training, experience and skills of the examiner. You can't get around it.