As always, check with the DM to see what's fitting; most have particular settings/races in mind and it's important to gel with that.
If you're allowed to pick from the kitchen sink, particular ancestries that fit:
1. Human. Open-minded as they come, and if Tsukiyo's an exotic god, a human could be just weird enough to serve him anyway.
2. Tengu. They love anything/one that's exotic, which means they're happy to interact with outcasts no one else would. Strangely, their personality is closer to Tsukiyo's gregarious past than his more quiet present, but still.
3. Anadi. They're wise enough to understand arachnophobia, yet friendly to make aims to overcome such instincts.
4. Goloma. They themselves are often misunderstood as scary monsters, so Tsukiyo would sympathize.
5. Kitsune. Not as strong as the others, but if your focus is on diplomacy, an ancestry with experience being all things to all people might be interesting.
Bonus: Tiefling heritage. Nothing like having demonic blood to lump you in with the demonized.
There's not much to build with a cloistered cleric at 1st level; just an ancestry feat. As for skills/background, anything to do with diplomacy or society would work. Being a voice for others means being persuasive, and society helps you to understand others.
If you want to make an *effective* cleric, you'd probably want to avoid any personality trait that's "quirky"; otherwise you run the risk of being dismissed as just another weirdo to ignore. Being someone others can't ignore, whether that's because of impeccable character, nobility, or competence, etc., would be crucial to winning a place to speak for others.
The domain really depends on what the character, per the feat description, is most interested in wrt to Tsukiyo. Putting the dead to rest? Spirit. Destroying wicked creatures of the night? Moon. Soothing the emotionally tormented? Repose. Delirium seems pretty contrary to Tsukiyo's purpose of...the exact opposite. Not sure what that one's doing there tbh. Having a midlife crisis? Change! Tsukiyo's familiar with entering a new phase of life.
Final note: It's not optimal, but Intelligence might be a score you go for. Again, the more languages and/or Society knowledge you have, the better you can reach the misunderstood.
While I'm just using the AoN description, I don't see anything about Tsukiyo being shy. Quiet, reserved, thoughtful, sure, but nothing about shyness.
