you need lots and lots of birch bark to make a wet wood fire.
not a survival situation for sure but on the spring canoe trip i had a lot of heavy rain.when it was over i gathered up a heap of wet to damp pine branches to make a fire.a bit of bark about the size of magazine page is the most i would use but this year i used it by the bushel basket full for a sure light fire.behind a couple camp sites i found down and rotten birch logs.with my hand ax i cut them open and shook out the pulp.at he fire pit i piled that up ,like knee high,and put the branches on top.the resulting blaze dried out the sticks which burned hot enough that i was able to add more and get a nice hot fire and not a smoldering punk.
lesson--take some time to do it right.