I was creating my inheriting form by creating a new form
public partial class NewDerivedForm : Form
and then manually going in and changing that to
public partial class NewDerivedForm: NewItemForm
The Monkey See Monkey Do answer to my problem is - Don't do that.
Using the Inheritance Picker template generates a form that does not seem to have the problems with the VS designer that I was experiencing. The auto-generated Designer.cs file created by the template is different from the 'normal' one, and the skeleton code created in the code file is different also. And, no .resx file is created at all.
I'd list the differences between the generated files, but then I'd have to explain them, and that is unfortunately above my current skill level.
Thanks for the help, guys, I really appreciate it.