I assume the SSN has been applied for. In my experience, I had a child born in a military hospital overseas. The SSN application was completed prior to the mother/child leaving the hospital. The SSN was received by us about 3 months later, give or take.
Here's another option. If you have already applied for the SSN, just wait, and if you get it prior to the April 18 filing deadline, you'll be fine.
If you do not receive it before the filing deadline and you want to e-file, go ahead and e-file without claiming the dependent. Then you can file an amended return to claim the dependent when you do receive the SSN card.
If you have not applied for an SSN card for the child yet, then do so ASAP and you very well may get it in time. I would expect there to be a consulate representative in the hospital, or somewhere on the base such as the legal office, which would be your starting point.
