This is a losing battle, and it's not worth your time to fight.
The people who are giving you a hard time for being 15 already have their minds made up about what 15 year-olds are like and what they can handle. Their preconceptions may be generally right with you being an outlier, or they could be wildly inaccurate. In any case, you're not going to change their long-held beliefs no matter how eloquent you are.
That's not your fault in any way; it's just how it works, especially on the internet.
So what do you do instead?
You're on the right track with this:
How do I convince them that my exact age isn't what matters, but the level of maturity that I display?
You've already demonstrated your maturity to them - they assume you're older than you are. You're right that your age doesn't matter, so don't let it even come up. You have several options when the subject of your age comes up:
1. Do not engage
If someone asks old you are, ignore the question. Act as though it was not asked. If you're in a thread with multiple people, respond to someone else. If the age-asker said multiple things, respond to the other parts of their message. By not acknowledging the question, you take a lot of their power away. If they keep harping on it, they begin to look like the immature one who is fixated on some irrelevant factor about you. The conversation stops being them telling you to prove yourself, and starts becoming them begging you to let them in on your personal life.
This is your best option, but it's the hardest. As a moderator, you know better than most that people on the internet can get under your skin, and they'll go out of their way to get a rise out of you.
Have you ever seen someone ask a politician a difficult question? The politician will respond, but you might notice that they answer a different question than what was asked. If you don't like a question, answer a different one. It's important that your response is still on-topic, though. Changing the topic would make it look like you're on the defensive, which suggests you have something to hide.
To borrow your example:
Me: mentions something about doing something IRL
Other person: Wait, how old are you? Isn't that like for kids?
Me: Ah well, we all have our guilty pleasures from our childhood don't we?
Hopefully they give up on their age question. If not, you'll have to move to another option, since deflecting too often makes you look defensive.
If you absolutely have to respond to a question about your age, acknowledge the question and indicate that it's not a question you're open to discuss. Here are some possible phrases:
That's not relevant.
The administrators vetted me before I became a mod. Here is how you can contact them if you have concerns.
I don't disclose identifying information online.
The trouble here is that it gives them an opportunity to press further. Be careful not to phrase your dismissal as a question ("How is my age relevant?"), as that invites further questions. If your dismissal doesn't work, follow with more dismissal ("I've already responded to that question.") or Do Not Engage.
Your goal is to never legitimize this discussion. They asked something off-topic, but inoffensive so ignore it if possible. Never provide evidence to justify your position. Never give them a number to work with. Provide contact information for a person to escalate to if they have a legitimate issue. Give your supervisor a heads up that this is happening, and that you'd prefer not to disclose your age.