As I (mis)understand it, I believe the issue may be caused by your having the outer bounds (you can see that with Use Preview Bounds ticked in the general preferences) of the artwork extend past whole pixel values. They should be contained within.
I ticked "Use Preview Bounds" and it didn't seem to make a difference.
I've uploaded four screenshots.
1. I start using Object/Artboards/Fit to Artwork Bounds. The logo artwork looks good.
2. I select View/Pixel Preview. As you can see, the pixels are now outside the bounds of the artboard. When I save as a PNG and place into a project, the logo then has a razor line crop across the bottom.
3. To resolve, I extend the bottom border of the artboard.
4. I deselect View/Pixel Preview. This is what I'm trying avoid having to do with every logo iteration.
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The artboard (and the artwork) needs to be aligned to the pixel grid.
And then when exporting "Use Artboards"
Thank you. I think the big key was checking "Use Artboards" on the export.
When you Use Preview Bounds it will make no difference to the appearance of the artwork, but you can use it to determine whether the artwork is contained within whole pixels as it should, as I said in the first post.
I believe that if you Use Preview Bounds and tick one of the bottommost reference points in the Transform palette and select the blue object, you will find a decimal/non whole Y value, corresponding to the fact that the bottom of the rounded part shown in 1 in the original set of images extends past whole pixel value of the Artboard border.
This means that if you Use Artboards on the Export without correcting the artwork, you are actually cropping off a bit of the rounding, giving it a flat bottom, which I consider a fundamental flaw, however little the flattening may be.
Aligning the artwork to the pixel grid has its limitations and may lead to changes in the artwork beyond your control (and maybe without your noticing it).
Therefore I believe the sound way is to correct the actual (raw) artwork to fit within whole pixels in exactly the way you want it, which Use Preview Bounds can help you with.
It may just be a slight move upwards in this case (depending on the unseen parts), in which case they would be no reason to change (or you may say slightly destroy) the artwork.