Ugh I feel your pain! Hard tile floors are no good some days!!!
I wear all black, and some days if my legs are bothering me, I'll wear my dress pants, and I have a pair of black Pumas that almost look like dancer shoes as opposed to real tennis shoes.
But for my flats, I've tried (and still haven't found perfect ones yet) doing heel inserts to help support my arch and the ones I tried helped a lot, but they bumped my foot up too high and then my shoe gave me a blister lol.
If you can get away with doing a more low-profile sneaker, try it, or even ask your boss if it would be ok and explain your situation. Better to ask and know than wait and wonder!