That's a lot of words you're using there buddy.
You need to optimise them there tokens 😂
So, to be clear, OpenAI OAuth, uses the Codex engine, not your Chat account. The Codex allocation is different, metered and capped, but your Chat account is not.
Why?
In your chat sessions, your usage is limited by your ability to think and type / talk, so it is not metered or capped.
In your Codex allocation (programming environment), you could prompt it with a single prompt like, "Rewrite all the software in the world using Fortran" and you would end up using 100% of OpenAI's multibillion dollar infrastructure and it would only cost you $20.
With an API key, you pay for what you use, so if you're Elon Musk with a trillion dollar credit limit on your credit card and no cap on your API spend, you could actually pay for and use OpenAI's entire GPU cluster, world wide 😂
As for web searches, you're not using OpenAI to do this, your Claw is searching the web locally. OpenAI is only providing the brain.
If you want to test to make certain web search is working locally for you, ask it to do some kind of real time search, for example "What is the current Bitcoin price?" Or "What are the latest headline news stories on CNN?" If that works you have web search, if it doesn't you don't.
If you want to see your Brave API search usage, you can find it here:
https://api-dashboard.search.brave.com/app/dashboard
If you want to see your OpenAI Codex usage, you can see that here:
https://chatgpt.com/codex/settings/usage
I only lightly use my Claw, yet my OpenAI OAuth Codex usage ends up using about 30 - 50% of my weekly allocation.