Daystrom Institute

Temporal Prime Directive typically means not messing with the past. I think it would be a bigger violation to bring them back to their original timeline depending on how much knowledge they had attained of the parallel future. I think your final point of Starfleet having no way to know the natural development of either timeline is the key here. There's a reason it's a Temporal Prime Directive and not a Multiversal Prime Directive, which is putting Starfleet in kind of a divine arbiter position of deciding what is the correct path of history for any given universe.

It feels like many things leak through weird anomalies, and parallel realities are shown and dismissed without a second thought. Like in Voyager, both Harry Kim and Naomi Wildman are technically from an alternate reality having crossed through the anomalies to the surviving ship's timeline. Everybody just kind of welcomed him with open arms while acknowledging that the universe is just a super strange place.

If the accident was an anomaly and not an experiment, Starfleet probably wouldn't really care. They would find them and determine how disruptive they may be through an interrogation with a full telepath Betazoid and let them loose on a Federation world with an equivalent lifestyle to their era and locale. Unless this was like some kind of potential intergalactic criminal, it's just some dude who deserves to live out their life as normally as possible given the situation. If it was an experiment, though, they probably would just confine the dude until a badmiral decided to try to have them recreate the universe-hopping device or otherwise consult them on how to safeguard against the intrusions.

I think the precedent would be the way that the Kelpians and Saru were treated with the Prime Directive. Saru having knowledge of what was beyond his planet essentially allowed him to be uplifted individually. If somebody crossed over similarly having knowledge beyond their planet in the same way could potentially also be uplifted and serve in Starfleet in some capacity, but my money is on anomaly dude just wanting to vibe somewhere chill. Otherwise, it's as dangerous as anybody from the Mirror timeline would be, and they typically defend against trans-universal attacks like that, so I would assume the fugitive would be considered hostile.