Based on your requirements, the solution I can think of is to use a browser extension that lets you create custom prompts that are injected on every conversation.
I’ve created one that lets you do that, but it is focused on voice, which I suspect isn’t what you’re looking for.
I strongly suspect that there are other that will do what you want.
I do want to warn you that there’s always some risks in using an extension, so double check that they only have access to openai’s web site.