I am looking for the best way (or at least any way) to trill between first space "A" and 4th line "D" on a Bb Clarinet. The passage where it is used is to be played softly, and using traditional fingerings is way too loud (especially since it crosses the register break).
Any ideas?

That actually isn’t called a trill; to be technically correct it is a tremolo. A trill is when you have rapid alternation of a note with the one above it.

Now for your question, I don`t think there is any easy way of fingering those two notes. Go to google and do a search for clarinet fingerings or tremolo fingerings and see if anything helpful comes up.