Conjure an IQ test and take it, obviously. My IQ when dreaming ranges from greenish-purple to twelveteen o'clock.

I once deliberately created and solved a quadratic equation in LD and concluded that my cognitive capacities are near my day-level at that moment. However, symbols tend to drift in dreams, and complex symbolic reasoning is impossible. Thus LD are better for new ideas generation.

Set up a webcam to observe your eyes. Use deliberate eye movements to record information and test whether your dream operates on the same time scale as reality. I understand that lucid dreaming is most stable when it involves vivid experiences, so a simple task that comes to mind is I pack my bag. With a computer and/or friend, you could see whether you can hear reality, and make this quite a bit more rigorous.

Edit: They did this in 1981. Eye movement works, sensory input doesn't. https://www.semanticscholar.org/paper/Lucid-dreaming-verified-by-volitional-communication-Berge-Nagel/459cf8dda2f68537e88dd7baa1d86dcc4b9febc7