True, you can't do this with GPT or Claude, at least not today. But you can do it with a local model: vLLM supports deterministic inference out of the box! (If you have a MacBook, you can confirm this yourself; just point your agent of choice at the site. It will download a small open model and run it with vLLM Metal.)
Naturally, I hope that OpenAI and Anthropic will one day offer deterministic inference; see the "Weights" page for what this might look like.