I have a piece that I attached a little extra piece of lace onto with a fabric glue.

It worked well to secure the two pieces of lace together, but it turned out stiff and detectable to touch. Good thing for me it was right above the ear where it isn't likely to be felt.

I'm thinking a softer, more pliable adhesive might work better. Something that is nice and flexible when it dries like the poly they use for hair units. I think ideally if it you are adding to lace you would sew the piece to the base, then use a tiny bit of glue to lock in the threads. My suturing skills are ok, but if I had the right kind of glue I think I could make a virtually undetectable AND nearly indestructible repair.