It looks like there is no way to show dim palette via API. The functionality of Dimension Palette only a custom UI holder for basic dimension settings. So it may be easier to reimplement the dim palette as custom form and display it programmatically.
Artem Taturevych, Application Engineer at Intercad (Australia)
translationXpert – add-in to translate SolidWorks models
That's unfurtunate. I was afraid that was the case. Making my own interface isn't a bad idea, but it would be too much work for something that pretty much already exists.