There is certainly increased evidence to suggest that the gut and brain are intimately linked. Sounds obvious to me. I think a substantial proportion of the body's seretonin is produced in the gut.
It makes sense that a healthy gut biome might help mental illness too.
It may be mediated by biochemical feedback that takes place between the gut and the nervous system and tied in to homeostasis governed by the endocrine system.
It's all linked.
Yes, I've heard of FMT. At the moment I'm experimenting with food/prebiotic/probiotic/medication balancing. FMT is a more extreme, quick fix way to re-populate an eviscerated gut biome. The theory appears sound and there does appear to be anecdotal evidence supporting it's application.
Good luck finding someone that will do it though!
Edit: I'd add this - if it did work, it would still be necessary to eat healthily, get lots of prebiotics and maintain a healthy gut flora. If one continued to eat junk, take antibiotics and have scans, there goes the neighbourhood (again).