Yeah, but humans don't inherit our parents gut microbiome from birth, for example

True, but those microbiota don't live inside our cells.

would an unintegrated internal symbiote of a single celled animal get split in half in mitosis? or would it just go on one side or another of the splitting cells?

my money's on the "it would go on one side or the other" hypothesis but this is a really interesting question! I imagine it could go the other way though if the symbiote is sensitive to its hosts reproductive signals and capable of replicating in time, so it probably depends on the specific host/symbiote pair