Steve,
Do I take is that you reference the date picker in the tiddler that you then export to a static tiddler? Static means a HTML only representation (No interaction or javascript) of what you see when the tiddlers wikitext is rendered.
In my view, In this case the date picker will not work in the static tiddler because its plugin code (perhaps also Javascript) is back in the source wiki plugin, thus I am not surprised it fails. The only way is to have a fully functional wiki or another javascript page.
Remember you could have a minimal wiki with only the tiddler you want displayed and any required plugins installed and provide a link to this seperate wiki.
Regards
Tony
You could use the html date picker but the value returned will be displayed but I doubt you can do anything else with it.