Just documenting this again, so I don't forget it, and it might help others too.
Specifically, how to make a PDF with transparency, using ImageMagick, so that it can be used as a stamp for PDFTK. This requires a PNG with transparency as input.
convert $INPUT.PNG -transparent white -background none $OUTPUT.PDF
pdftk $FORM.PDF stamp $OUTPUT.PDF output $COMPLETED_FORM.PDF