I am a passport person, and I know that passports CAN take a nice, quick time. But in your case I doubt that even I would send off for a passport. Given that you don't live where you were born and there's a potential complication of it being a Naval BC (didn't even know there was a separate thing unless you were born outside of the US?), there's no knowledge of how long it might take to get a second certified BC.
Now if you googled around today and maybe called around and found out that you could get replacements for everyone ASAP, then I might do it. In my case, I know we could drive over and get certified copies of DH and DS's inside an hour or two. For me I could have my dad get one b/c he still lives in the state where I was born, then he could send it up to me. Easy. If it's not easy for you, it's different.
Other than that, I would take *this* trip, and then get the passports for the future once you're home.
If you have an emergency while you're gone and have to fly home...they will deal with the passport issues at that time. It will likely be more expensive than even expediting them now, but that might be more OK with you should it be necessary.
ETA....I was just googling... You said it's a naval BC but you were born IN Virginia. From what I skimmed (search term "naval birth certificate virginia") it seems that those are simply filed with the Registrar's offices in VA, so you might simply be able to request that. The main vital records place is here:
http://www.vdh.state.va.us/vital_records/ They use vitalchek for their out of state requests. Might not be as hard as you thought. What seems complicated is when you're born out of the country.