You can pour under water, but not on mud. If you over excavate, perhaps you could try pouring a mud slab? This is low strength concrete that doesn't have a structural strength, but is intended to provide a solid base for the structural concrete. Being concrete, it won't shift and settle like gravel which would be my concern it being under a footing. All you need is 2,000 psf (vs 4,000 psi for structural concrete) to provide residential bearing capacity. The mud slab doesn't need a particular thickness but it should be 2-3x the width of the footing. The big advantage is that once the concrete sets, you'll have a nice surface to set footing forms on top of. Often lean concrete is used for this purpose (600 psi), but I'd suggest using 3,500 psi concrete because the excess water will weaken it.
I will say that your photos don't show much for mud. The excavator would only need to scrape off a few inches to get to final grade. If this can be done the morning they start setting forms and can stay off the footing footprint you might be ok.
I agree, you are probably dealing with a perched water table. Because you have a walk-out basement, I'd suggest directing the drain tile to daylight so it can work continuously.