So you have a reasonably readable dialogue here, with none of the truly devastating pitfalls I see all the time around here like mixed speaker paragraphs, or avoiding ‘said’. Presumably this means you can fix any little mistakes so I won’t bring those few up, and focus on content.

It’s a bit hard to believe, bordering on confusing and unreal for me. Now some of that low level confusion clearly has answers, like how Ron starts calling one of the many named characters brother in the last few pages despite having said he’s an only child. I can tell there’s more context there, but that it’s not here. That’s fine.

What is less fine starts with Emron’s attitude, extends to his presumable goal, and then stretches out for the entirety of the dialogue. So attitude first. Emron’s very presumptuous from the start, and not in a nine year-old way (in fact I don’t think he sounds like a nine year-old for a second, not in dialogue or in the few internal thoughts). He hesitates for only a moment to reveal a birth right that absolutely will get him no favours, and he seems to understand that but do it anyway. He’s rude and that culminates in an explicit (and stupid) insult to a shared blood relative, the only tie to these people by blood in fact. Arguably he should be doing the exact opposite, praising the shared relation and begging for mutual support and beneficence based on some combination of relation and the blood tie’s own good nature.

This combination of excessive maturity for his age combined with a bad attitude made me tired of Ron in three pages. He’s clearly been here a while, and built relationships, but he’s this rude to the powers that hold his life in their hands? At nine? Ridiculous. He’s lucky he wasn’t killed out of hand for the fact he’s been too-close friends with the High Prince’s daughter under false pretences, no matter what argument he could make; not actually making an argument for a lengthy period except umbrage/insults/rudeness would seal his fate as far as I’m concerned, between the political threat he represents and all the above.

But let’s say you refine how he behaves and asks for things and go to his goal. What is his goal in this dialogue? There isn’t enough internality exploring this. His setbacks and risks are represented by a gross metaphor, but his realignments/compromise/desperation, even his initial goal, aren’t explored internally to any serious extent. Is he really trying to go south? Is it really to petition an uncle he’s never met, whom no amount of renouncing would actually make feel safe about Ron existing, on the off chance the uncle wouldn’t kill him? And is it a compromise to seek service instead at the end? How much of one? I don’t know and can’t tell what Ron’s thoughts are on the matter, and yet this pendulum swing, I would argue, is the most important part of this dialogue.

I’d suggest you work on both of these. Without much greater leverage somewhere for Ron I’d start by making his strategy unceasingly polite and appeasing. As you show, he already has people in this very court who will want him dead for existing, he doesn’t need to antagonize the ruler or speak rudely on top of that. But take a critical eye to the whole dialogue based on the whole critique.

Side note, I’m getting too heavy A Song of Ice and Fire vibes, and imo you’d benefit from breaking away from it a bit. The House thing is understandably seductive, so maybe escape terminology like banners/bannermen with your own take (or the simple ‘vassals’). Also if you want Ron to be likeable, imo making him a light misogynist in how he reduces heartfelt pleas for his safety from women (some of whom matter to him) to a him-oriented manly ‘better not hide behind women’s skirts’ moment is not the way to do it. I cringed. It is in fact possible to be appreciative of others’ concern and yet seek personal resolution without being a dick about it in your personal thoughts. Having insufficient internality about the compromise turn and what Ron’s goal(s) and aim were here, but getting a sentence where he basically encourages himself to be bold by denigrating the efforts of women, was an especially potent combo to help me dislike him. I know many readers wouldn’t even register that, but it’s there, it unnecessarily risks turning off particularly women readers, and it especially is a weak way to portray strength or courage and a goal turn compared to a proper, personally motivated, exploration of same.

• I like the dialogue, but the some of the descriptions of what's going on overdo things a bit, both due to their old-fashioned language and how long they are. Old-fashioned speech is fine, but "droll" especially was immersion breaking for me.
• Some of your descriptions of how your characters speak are fun and memorable. I like Vinik slipping on seriousness, right after seeming constipated.
• I would try to reign in the descriptions in last few paragraphs. The non-speaking questions and descriptions go on a bit long, and drain some momentum out of what has been a pretty fast moving page.

Emron entreats the king(?) not to see him as a rival for the throne. The king demands Emron to "denounce" his father. Wouldn't it make more sense to swear loyalty to the king instead?

It's clear you have picked up a lot from books and TV shows, but your prose is just stuff you've read and heard tossed together. You remember the phrases in isolation, but you don't employ them in the service of a coherent story. There is no synthesis. It's a salad.