Don't use any commercial product.
I presume the dress is white, if not,
do not do the following.
Also, is the lace cotton, silk, polyblend?
Do not do the following if it is silk.
Polyblend is hard to figure unless you know
exactly what the blend is.Try ONLY a
Get a bottle of 100% lemon juice.
Try this first on a tiny, tiny area.
Squirt lemon juice on the mildew.
Put dress outdoors in the sunshine for a few hours,
moving as necessary to keep the mildew in the sunlight.
If it doesn' work on the tiny area, forget this method.
Get some denture cleaning tablets, dissolve in water.
Dip a tiny piece of the lace into the water.
Let sit and see if this removes it.
If these do not remove the mildew,
take it to a very, very good dry cleaner as soon as possible. The longer it sits on the lace, the harder it is to remove.
You don't want to damage something so precious.