You actually say that every case "works as expected" so; we really can't tell where you are getting an error. I have lots of pages with both Selects (although never double-click) and a non-classic date picker.
Moreover, it would be helpful to know, exactly what is going wrong when it doesn't work as expected.
I have a region with a Select List Item and a non-classic Date Picker Item.
The Select List requires a double-click to work. Not a single-click.
1. Remove the non-classic Date Picker the Select List works as expected.
2. Add a classic Date Picker the Select List works as expected.
3. Combine a Select List Item with a non-classic Date Picker Item in 4.2 the Select List works as expected.
It seems to be something to do with the non-classic. Maybe a Java confict?
Is this a known issue? Is there a work-around?
Select List. Non-classic Date Picker. Same region. It does not work. Requires double-click. Single click does not work.