Concern: you mention it supports the actual ChatGPT Web API -- isn't this against their terms of service? I'd be wary of publishing something that hits internal APIs.
Suggestion: I would rename what you call the "ChatGPT" (non-web) part of the backend to "OpenAI API" instead. There is already enough confusion in the world between ChatGPT and API access to the underlying GPT models.
There seems to be two extensions, one called "chatgpt" and one called "chatgpt-web" (https://github.com/chieapp/chie/tree/main/src/extensions). The latter seems to be using https://chat.openai.com/backend-api/conversation which is OpenAI's private API for ChatGPT. Using this (as a user) seems to be able to get you banned, as far as I can tell by some reddit posts as of late.
However, the extension called "chatgpt" seems to correctly use the right endpoint: https://api.openai.com/v1/chat/completions, but it's not ChatGPT anymore when you use it, so the naming is a bit confusing as you said.