I'm not sure if this solution will work for you, but here goes:
The problem is that the properties inside the Font are read-only. The solution is to create a new font, like this:
btn1.DropDown.Font = new Font(btn1.DropDown.Font, FontStyle.Bold);
Source[
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Another potential way to do this is:
This One[
^]. You could perhaps use the concepts in this solution to apply against your selected item.