How to avoid error "Constructor on type 'MyType' not found" when inheriting a base class

That is completely impossible.

The form you see in the design view is an actual instance of your base class.
If there is not default constructor, the designer cannot create that instance.

You can mark the constructor with the [Obsolete("Designer only", true)], and make it throw an exception if called when not in the designer, to prevent other people from calling it.