You can create the dot pattern by defining a scatter brush with a small circle. Apply this to a larger circle, then adjust the scatter brush spacing so that the first and last item overlap (using the up/down cursor key to adjust the spacing value will help. After you have one cirlce of dots you will want to use duplicate larger rings of dots and rotate 45 degrees (transform each can make this process easier).
You should now have a symetrical pattern of circles looking like a halftone pattern. You can then use the bloat tool, or open this in photoshop adn use liquify. The distorting part of this may not be as much control as you like, but hopefully this post gives you some ideas towards the correct direction to proceed.
Would be nice if the symbol sprayer could spray symetrically as I liked your original idea, or if you could atleast use the symbol distortion tool on brushes.