Personally i wouldn't touch it.If it's that fragile,it will do nothing but frustrate you!! I'd tell them sorry,but it's too old and wouldn't be worth the time,money and energy spent to try and save it,and show them how fragile it is.
JMO
Kathi

I cut a small piece of the satin to test it by washing it in my washing machine on a gentle cycle with some bleach. It cleaned a small amount but held together and didn't fall apart. The lace is dry rotted, so I won't be using it. THe lace seems to be fine, other then being yellow.

I thought my only options would be:

#1 Not use it at all
#2 Use it and just have a yellow baptism gown
#3 Use the yellow lace and cover it with a white lace or organza, etc.

Since it passed the wash test without falling apart, try either Rit Color Remover or Clorox for colors and see if the yellow at least evens out and fades to some extent. If it does, then you can decide which option would look best with your new shade of yellow and or white. If not, then option 1 was the only remaining choice so nothing is lost.

The product that never worked for me, but others swear by it:Oxy Clean.

Though it's a wonderful idea I'd be very concerned about putting something that old on a baby since you mentioned mildew. I'd ask a pediatrician before possibly exposing a baby to serious illness.