Your main issues are exposure control, white balance, and keeping background light off the subject.

To get a white background, set exposure deliberately rather than trusting automatic metering. A scene with lots of white often fools the meter, so you may need to increase exposure. Also set white balance manually for a neutral result, or correct it later in Lightroom if you shoot RAW.

For the background itself, increase the background lighting while controlling spill so it doesn't hit the subject. If the side lights make the subject clip before the background reaches white, that means the subject and background need to be lit more independently.

As for the clipping warning: yes, the camera warning is typically based on the embedded JPEG preview, not the full RAW data. That preview reflects the camera's current JPEG rendering settings, including white balance and contrast, so it can show highlight clipping even when the RAW file still has recoverable detail.

In practice: shoot RAW, set white balance intentionally, expose a bit brighter for the white background, and adjust the final balance/brightness in Lightroom if needed.