James,
There is just no way if you are giving yourself only a week. The country is narrow, but very long, you will be spending more time on the road than anything else. On top of it, you want to see the country AND go birding, that is ambitous to say the least.
If you have two weeks, you could try the following (still rushed, but better than nothing):
Day 1: Arrive in Ha Noi, sightseeing depending on ETA
Day 2: Visit the Museum of Ehtnology, Temple of Literature, One-Piller Pagoda, HCMC Mausoleum.
Day 3: Have a car pick you up real early and head for Cuc Phuong National Park (book ahead).
Day 4 and 5: Bird Cuc Phuong and head back to Ha Noi.
Day 6: Get an early flight to Da Nang and head for Bach Ma National Park.
Day 7: Bird the Park and head for Hoi An
Day 8 and 9: Explore Hoi An and head for HCMC the last day.
Day 10: Take a flight to Da Lat and start birding
Day 11 and 12: Bird Da Lat and go back to HCMC
Day 13 and 14: Explore HCMC and fly back.
I wouldn't go as far and say that the country isn't set up for birding, it is just a little harder than elsewhere.
If you want, and if your budget permits, I could set you up with a friend who runs tours here in Vietnam. An English bloke, he has lived and birded here for ages.