Enter grey thread! It seems like magic, but it blends seamlessly with both bright white lace and lavender fabric. Here it is on the lace side--there's a large zigzag on both the top and bottom of the lace, but you can't see it at all.
And here it is from the back.
Mettler Meterosene color 0854, also called "Saturn."
Grey thread works for all sorts of lighter-colored fabrics. Give it a try!