Ok, we're going off at a tangent here. I didn't suggest the Projection Deformer at any point; only that we use the project to surface function within the Fit Circle tool.
But I did you give you some slightly incorrect advice at that point - I had forgotten that this 'project to surface' function shouldn't need an additional object to project to - I imagine I initially thought it did because because your holes were pre-existent before you tried to round them, and I wasn't sure it could still predict what the absent surface was doing. However, in subsequent tests today I have confirmed we don't need an additional object for Project to Surface to work in Fit Circle. In poly mode, with the hole polys pre-filled that works reliably.
With that said, the way you are wanting to go about this is still not the best way. I'll say it one more time in an attempt to save you the work, but trying to get topologically perfect circles, in that pattern, on a standard sphere, presumably planning on using SDS to negate your relatively low polycount (?) there is a very high chance you will never be able to get a fully perfect sphere and perfectly rounded holes. You will always be able to detect a subtle pinching or concavity around the holes because if you choose this approach it is simply not possible to keep polygon spacing even enough to avoid them !
But by all means progress with your current methods and find out for yourself ! 🙂
To give yourself the best chance of this working I would suggest that you put a spherify deformer in a null with the parent SDS of your mesh. Then at least it can work with the SDS result where the polygons will be more even and a lot more of them, rather than with the base mesh directly. Again, you need to do this before any thickness is added.
CBR