How dry / wet is the soil? I assume with clay the soil can be water-logged, but I don't know how much rain you get. What weeds grow there currently? That might be a good indicator of what will thrive there (some weeds do better with really moist conditions, other in really dry conditions).
How shaded is it under the oak tree? Is it like a single tree in a field (relatively high light), or heavily shaded with
trees all around?
I could imagine dutch white clover forming a low and low-maintenance ground cover, but I think it might want
enough light.
Violets are a great suggestion, I find them in my wet clay soils in shade (I'm in east TN in the Great Appalachian Valley, technically a rain forest). Violet seeds are not viable for long, but they grow commonly as a weed so I would just look to dig up violets from somewhere else and transplant them under the oak to get them started.
Comfrey isn't low-growing but will grow back readily from being mowed and will shade out other weeds and will provide lots of nutritious mulch when mowed (assuming you are mowing and that's why you want low-growing?), it's a common
permaculture trick to put comfrey under the canopy of trees and to treat them as a chop and drop nutrient pump.