Have you looked at the Behaviours of the field. If you have Select Entire Contents of the field checked it will behave differently...
Perhaps a pause built into your script might allow the keyboard to load first,
What I discovered is that on the iPad, the field MUST be on layout in order for the field to remain active with the date picker displayed. This is unlike the iPhone, where the date picker remains active even if the field is off layout. Attempts to move the field so that it was barely on the layout (by 1 point) did not work. So I ended up shrinking the field width to 1 point, and moving the field onto the layout in an inconspicuous place -which hopefully I will always be able to find later, as it's nearly invisible in layout mode! When in browse mode you can still see the selection box around the field when it is activated, but it is so small that it is not as visually distracting as it was before. I can live with it.
Interestingly, I found that the iPad keyboard MUST be displayed when the date picker is displayed, not sure if that is an iOS thing, or a filemaker thing. Odd...