If you add a stroke to the font Shape marker, the stroke will expand from the character edges, obscuring the character shape. This is close, not ideal...
So what I do is duplicate the Shape marker, add the stroke the duplicate, and move it below the original character. So while there isn't a specific Halo property of the Shape marker, this creates the Halo effect.
Turns out if I just use "Text Marker", I can do it. But this is a zillion more clicks than it used to take with ArcMap.
I may be misunderstanding the question, and I can't view your attached images. Or maybe this was added to ArcGIS Pro after you asked the question. This works for me (this point symbol is from the "ESRI Default Marker" font): Click the symbol to open the Symbology pane, make sure you're on the Properties tab, click the Symbol (paint brush) icon, and you should see options for Halo:
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