Um... how about keeping it simple and use a 35mm something? Hobby side of photography aside, you'll be working in constrained environment. Small and quiet 35mm may be the best gear to use. If you don't have one, one can be purchased very inexpensively. Maybe a small and slim tripod will be permitted for a quick use? You'll be shooting through two layers of plexiglass anyway, so quality is compromised already.
Congratulations and I'm so glad she'll going to make it. I'm sure she is a beautiful baby!
Does Platinum printing involve a contact print? I wouldn't be happy sending out a contact print of a 35mm negative.