Yes, bitcoin core fits well with what I have described, but I would like the user to have the opportunity to use other currencies(4-5 popular currency ada, eth, usdc), unfortunately, the specifics of the my site do not allow using the payments services(coinpayments or btcpayserver) that you mentioned. at the moment I'm leaning towards binance with its ability to create sub-accounts in a business account via api.
I don't understand what is the point of keeping coins on centralized exchanges when you don't really own them and real owners can freeze your accounts and transactions at any point...it's not conspirasy theory, it's happening all the time.
If you want to havce security theater you can just create simple website database of numbers and pretend you are transacting between users.
There are wallets available for multiple currencies and you can just use them, shitcoins usually have low transactions fees, and most of them are worthless centralized junk.
One open source mobile wallet you can check out is called Unstoppable.money, and it might be used for your needs.
You could also use peer-to-peer marketplaces like LocalCryptos.