Ooh this is such a great question...I've been so focused on what he would look like that I haven't even begun to explore what he'd actually be like as a Harbinger.
I think it would take a lot of work, but he's probably just fall into the same role/rank as Arlecchino - she's grooming him so specifically, and she might even offer him up as a replacement if she wanted to retire imo.
I think he'd struggle a lot with it, as well. He'd take on the role of Father for the Hearth quite naturally; taking care of vulnerable kids would be second nature. But I think he'd have some serious reservations about the Tsaritsa and everything involved in being (essentially) one of her lieutenants. I just don't see him being able to convince himself to believe in The Cause (like Childe), or really lean into using the position for his own gain like many of the others. For him it would be a necessary evil to protect and fund the Hearth, and not much else.
Unfortunately I do believe he'd be incredible at the job. He's already got all the characteristics of a great diplomat - charisma, intelligence, the ability to read people. So I imagine he'd quickly become a favorite of the Tsaritsa and the main face/representative of the Harbingers.
As for relationships, I think Freminet and Lynette would have similar reservations. Lynette seems like she'd be more able to rationalize it all as a necessary evil to take care of the kids, though, so she'd probably be the one Lyney went to when he needed someone to reinforce that he was doing the best/only thing he could do. Ultimately it might strain things a little with Fremi, though.
Wriothesley...would get it, but absolutely hate it. He's the poster child for working within The System to fix it/make the best of it, but at the same time, being a Harbinger is on a completely different scale from being the Warden. Everything Lyney does has the potential to have a massive international impact and I think that would scare the hell out of him - especially when it could easily make Lyney a target for all sorts of plots and political intrigue.
I think the Wrioney situation would be affected by what stage their relationship was in when Lyney became a Harbinger ultimately. If it happened while they were in an established relationship, there would probably be discussion of how he might get out/Wriothesley would bring up his concerns openly and offer a counterpoint to Lynette's more pragmatic approach. But if they hadn't discussed their feelings...well I feel like I could write a horribly angsty fic about that tbh 👀 It would create such a conflict of interest and put even more distance between them...

Thank you for the prompt, thinking this through "out loud" was fun!