I don’t know if you see a Pdoc or not but... I had a foster dog that naturally alerted me to seizures. I talked to Pdoc about it and she wrote me a prescription for him. It did not make him a service dog but it did give him access to more places. I did start sending him to training to be a service dog but then we found out the seizures were a weird reaction to a medication so we just changed the medication.
Also, there are PTSD service dogs as opposed to them just being considered emotional support because it is mental health. As someone who has needed service animals the biggest problem with people passing pets off as service animals is the lack of training of the dog and the owners lack of understand of the rules for service animals. I have been attacked twice by dogs wearing service dog vests. Thankfully my service dog blocked the more aggressive of the two.
I also understand the not wanting to make T tell you no or even risk T having to tell you no... even if you are OK with no as an answer. My T tries very hard not to have to tell me no because I struggle so much with feeling like I am too much or a burden. I am thankful that when talking to me about boundaries my T tells me lists of what hi CAN do. Even if you can’t ask about bringing your dog yet maybe you can talk to your T about how hard it is for you to ask for what you need from them. I am sure if you are like me there is more than just the dog that you would like to ask for.