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If a wooden rosette is large enough - say a 6" ring that is about 3/4" thick and about 1/16" in height after sanding/scraping flush - could it have an effect on the stiffness of the soundboard in that area?

These are the ball park specs for the rosewood rosette in the Carpathian top on my current build that is about .105" thick at the rim of the sound hole but tapers out to about .098" at the edges. From what I am told, I need to use an oversized spruce bridge plate (with a rosewood patch for the ball ends) but I don't want this one to sound like a 1970s Gibson if I can help it!

The sound board thickness seems OK to me. I do not think the rosette will make much difference at all. You are just replacing one wood for another and it is surrounding a 3.75" or so hole. If it is a steel string I would just put in a normal sized maple or rosewood bridge plate.

from my experience what happens is the grain direction of the rosette to the top may have issues through typical shrinkage.
Most time the rosette cracks before the top. I find the tile style holds up better but again wood will shrink and expand.