Armed with a new circle-cutting tool, I thought it would be smooth sailing cutting out the balloons and the outline rings.  Nope.  There were two things that made this challenging:  the fragility of the oak rings where that grain runs radially, and the imprecision of the cutting tool.

The tool I ordered has a trammel point and a grippy foot that slips over the trammel point (to avoid making a hole in the material).  The order of operations I tried first was:

The underside had been taped where the ring would be, so the ring held together.  An earlier attempt without tape was a bit of a disaster and the fragile ring broke apart.

Unfortunately, due to a slight tool wobble when cutting with the trammel, the resulting ring was oversized and wouldn't fit right.  I started over, trying to reset the tool to the outer radius.  The tool does have markings on the side, but they're not very precise.  I adjusted as best I could and cut another oak disc.  It fit into the white field, but there were irregular gaps (tool wobble).

I decided to keep going just to work out a technique that would give decent results.  I taped the disc in place, reduced the cutter radius, and cut the inner ring edge using the trammel point.  Not bad results, but there were still gaps and the inner circle was very slightly off-center.  But, the order of operations seemed to work.

Using the same steps, I cut the smaller circle.  I went with the smallest radius the tool could cut (which is larger than the original design).
This went well, except that I made the ring width larger than I intended to. 

The smaller ring looked off compared to the larger yellow circle, so I went back and re-made the larger ring and disc (since it had gaps anyway).  Here I've started back-filling the lines, which went smoothly.  It involved cutting out little pieces of the white background and fitting-in the thin strips.

Here's when I spotted my first goof on the pattern.  The circled white square was missing, and I added it.  This shifted things down one square and meant I needed to re-fit and extend the lower right corner of the the balloon module.  The lower left would be cut out too, but that had already been the plan.

That wrapped up the left balloon module, and it could be taped in place.  Next, I'll start filling-in to the left, as I've already started here:

Oh, and the 2nd goof I found was the upper left square in the photo above.  I had placed a black square instead of a blue one.  And that was the very first square I had laid down on day one.  Not a very good start.  😁 
The repair was simple using an Xacto knife.