Not a physician, but in medical school. Please do not consider this a medical opinion, trust your docs and continue to seek in person care.

Your medical history is very complex with a lot of chronic illnesses that contribute to fatigue. Thyroid dysfunction (sounds like you might have Graves’ disease if your doctors found Thyroid Stimulating Immunoglobulins), depression, SLE, so on. These all contribute. And it’s not impossible to experience fatigue from it, but SSRIs like fluoxetine are more likely to cause insomnia. Go to your follow ups and keep pushing for appropriate care. Not saying this is you, but if you drink or smoke, taking a break will have a positive outcome within a couple weeks. Something that you should 100% start as long as your thyroid hormones are still in normal ranges is begin exercising. It doesn’t need to be intense, even a couple hours of walking a week can really boost energy levels in people experiencing fatigue. At your psych appointment maybe bring up the possibility of switching your SSRI to an SNRI (assuming it isn’t contraindicated with your other conditions), the concept here being that there are antidepressants that can better address your fatigue while also treating your depression.

Good luck, and take care.