Fine looking project.
I would probably cut a backer out of a darker wood (or wood stained darker than the Baltic birch) and
leave the original cutting as is. You could glue it to a square piece as shown in your original post of
cut the backer round. If cut round, I would probably make the circle about 1/4" to 1.2" larger than the
over-lay.
Of course, this is just MY suggestion. Different people have different tastes.
You did a fine job. Thanks for sharing and God Bless! Spirithorse