If you call the Focus method of RadAutoSuggestBox, the focus doesn't propagate to the child RadWatermarkTextBox.
To work this around, get the RadWatermarkTextBox control and call its Focus method instead.
var textBox = this.autoSuggestBox.FindChildByType<RadWatermarkTextBox>();
textBox .Focus();
// Note that this code should be executed after the RadAutoSuggestBox control is loaded (on the Loaded event or later).