yup, that'd work for just one datepicker control on the page. I didn't really read higher up the post. I can see you have multiple datepickers. All you have to do is add this prefix
code
$('.Screen1_mobiledatepicker_5 .datePickerControls .datepickeropenbutton').trigger('click');
/code
OR
code
// The name of my page is "Screen1"
// And the name of my datepicker is "mobiledatepicker_5"
$('.Screen1_mobiledatepicker_5 .datePickerControls .datepickeropenbutton').click();
/code
So now you can see how that name was derived. Alternatively you can just look that up in the console.
I used that in my datepicker "Tap" event.