I would fully expect that a monotonous diet leads to a heavy skew in the gut microbiome as specific bacterial species that thrive on that diet are selected for, others against. It makes some sense that a fecal transplant could repair the damage. If the diet has shifted or expanded, the transplant could lead to long term benefits by restoring newly-viable bacterial species, perhaps by facilitating digestion of the new types of food.
I’d be curious to see a factoring out of the diet composition, gut microbiome, genetics, and severity of autism symptoms.