Implementing peer-to-peer transactions question

Hey guys, I been working on a project for 2 yrs now. I wanted to make a ERC-20 token for my project and give the initial users free sign-up tokens. Then later, these same users can do peer-to-peer transactions within the app.

Problem is, cost of transaction is too high where the value of the tokens(ERC-20) is almost none(initially). I see there are solutions such as Layer2 scaling like Arbitrum, optimism, loopring etc..

Real question is, can I do my own transactions and post these transactions to the ethereum ledger? Without depending on known L2 solutions.

So when users peer-peer transfer within the app, I can handle the transaction for them in the back-end server and post the transaction to the eth blockchain.

I see you can host your own node, would that be the solution? Just accept connection from my back-end for transactions? Is that enough or its not a possibility?

tldr; I am trying to avoid high transaction fees but want to make independent solution