@Snowder The first option would be the way to go for now. The sample ChatGPT-style web app implements this using CosmosDB, which you could use for reference.
This is a similar default for most APIs within Azure, where in the request/response payloads are not usually logged. This is for both privacy of users and freedom to integrate in whatever way developers need to.
While there is no feature like it today, feel free to raise a feature request to it here with your scenario for the team to assess it.