Is there some way to take advantage of the fact that I own 10 BTC without spending them or giving up control of them? Is there some kind of "Bitcoin society" where I can instruct my Bitcoin Core to run a command which then spits out some kind of cryptographic code which verifies that I control/own the number of BTC in my wallet?

And then I give them that code, which only proves that I (as in "whatevernickname123 on BitcoinGentlemanclub.com") own this amount of Bitcoin, so that my account there can be labeled with a "verified to own X BTC" text?

And then that, in itself, without me having to try to convince anyone with words or easily faked screenshots, gives me power of the same kind that somebody who is known to own X amount of US dollars on a fiat bank would have without actually having to spend that money?

I don't want this in any way associated with my actual name/person; only the account on this site which I assume exists and whose purpose is to connect people who own Bitcoin for potential cooperation/business deals, avoiding all the scammers because they have to cryptographically verify that they own the coins.
Bitcoin Core > File > Sign message, then you can sign your nickname. Be careful, this signs for you address, not your coins, and anyone who can use a blockchain explorer can see to whom you send if you ever spend your coins.