No. Adding soil like that can harm the tree by smothering the root and rotting the base of the tree. Edging is ok so long as the mulch ring grows as the tree does. Not only that, anything you're thinking about planting in there is going to have to contend with the birch roots. It's a losing battle. I have some neighbors that thought the 10" block from lowes around their trees looked good. I guess it's subjective but their trees are declinging and I'm waiting for the hoa to send them a letter about the whole thing.

I'd avoid any edging (below the surface) that close to a tree also.

What you are talking about is called a fill. Cuts and fills are often detrimental when undertaken under trees. However, since you are on sand and the tree is a riparian species a shallow layer of highly aerated material like potting soil might not bother it. But potting soil placed over sand in a hot climate might disappear pretty quick. And the tree would look better without a ring around it.

Trees damaged by fills don't necessarily suffocate or rot, in a particular instance suddenly reduced access to seemingly incidental moisture from showers and light watering may have a serious impact. Trees in humid conditions often root just below the surface, with roots sometimes even popping out of the ground. When a fill is undertaken such roots are no longer supplying the tree with the water encountered there. On shallow soils such as are often found under forests here there may be a thin layer of organic debris and roots on top of an infinity of mineral soil and rock, with only a small percentage of anchorage roots penetrating a bit into the subsoil. Even tall trees, when failing and falling are seen to have a pancake root system that looks beneath the base of the trunk like the tip of a bullet shot against an impenetrable surface.

Trees in humid conditions often root just below the surface, with roots sometimes even popping out of the ground.

That describes this tree exactly.

Alright, scratch that idea. This tree is doing too well the way it is (about 17' right now)to risk making it worse over something like this.

Thanks for all the information, and suggestions.

That's good to hear. River birch that far south is not very common. You want to give it the best chance to succeed.