In my experience, it wasn't so much that there are separate tests for people with hearing loss only or people with tinnitus, as it was having an audiologist who conducts the test knowing you have tinnitus and the knowledge to help you with both your hearing, if necessary, and your tinnitus.
My hearing loss, while I was aware of it, really wasn't enough to issue me hearing aids. I do have high frequency loss in both ears but I can hear good enough without the aids. The problem was I couldn't hear good enough over the tinnitus and also one of the hearing tests indicated that my speech recognition was less than 80%.
So, having an audiologist who understood tinnitus was good for me because she recognized that it was the tinnitus that was affecting my hearing more than actual hearing loss. Knowing this, she has been able to issue me hearing aids that improve my speech recognition and attempt to give me some relief of my tinnitus.
I've been dealing with tinnitus for 50 years and it has gotten severe about 10-12 years ago. I understand your suffering. Audiologist tend to get a bad rap here and it very well may be warranted. I've been very fortunate to have an audiologist who understands and is very knowledgeable of tinnitus. She keeps trying to help me even when I give up. So, to answer your question, it's not so much a tinnitus specific test as much as it is an audiologist who will test appropriately for your tinnitus and hearing loss. This will include very high frequency tests, speech recognition tests, and even tests to determine your type of tinnitus. By that, I mean if it's tonal and what frequency or frequencies your tinnitus is.
Take care and I wish you well.
Bobby