to be clear: are you wanting the UI to show all matching items in a list
beneath the line edit, or are you wanting it to just complete the "best" match
in-line?

I'd like to complete the best match in-line, either as an autocompletion in the LineEdit itself (like in the KRunner text field) or as a dropdown, but I guess that it's not possible with the current API? So would the second best solution be to present a list beneath the LineEdit? If so, which Plasma UI elements are suitable for this?

That's how it looks btw: http://imagebin.ca/view/26BfLjo.html