I'll start by stating the obvious, if you can switch clinics or your psychiatrist, do so. Just like with any medical specialty there are good and bad practices and good and bad physicians. I have experience with both, but can guarantee that are some great psychiatrists out there.
The fundamental problem with the field, in my opinion, is diagnostics sucks and the treatment options suck, for the most part, with extremely limited efficacy. There is no blood test or imaging for diagnostics. It's an art form, and this is why so many people are misdiagnosed by psychiatry. Regarding treatment options, it's, at best, akin to psychiatrists are only capable of bringing a knife to a gun fight. Outcomes are typically dismal. This is not the fault of psychiatrists, per se, but it underscores how little is known about the human mind.
Next, your discussing lithium and it's efficacy with depression. Lithium is more effective with mania vs depression. In fact, the largest study ever on depression, the Star D study, concluded that lithium was LESS effective than placebo in treating depression. This is a main reason why psychiatrists oftentimes prescribe Lamictal along with lithium, as Lamictal is shown to work better with depression, but not as well with mania.
Getting treatment from a vet is a bad idea, if in fact you are not being tounge in cheek.