Is there a way to add custom UI - for example a dropdown - to the view container title, similar to how the Debug view container shows a dropdown of launch.json configurations?

No, that's not possible. Visual Studio code does not support any standard HTML to be used in the parts of it that can be extended by an extension. Webviews would allo such kind of UI, but they cannot be used as general UI extension, only as different document type.
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