Just sign up for their API and use chatboxai.app (free software), use the GPT4 API code, and you'll use the model through an alternative front-end for pennies on the dollar.
I'm using it extensively and rarely spend more than 1,20 - 1,50 a month.
If they want proper anonymity, the user needs to protect against fingerprinting from the duckduckgo website (Tor or Mullvad). If by anonymity you are meaning from OpenAI, then duckduckgo needs to be running user's text prompts through a paraphrasing LLM to normalize text and avoid deanonimization using writing-style Fingerprinting.