This is a very wet gingerbread batter.
Should I substitute the applesauce for (some portion of) the eggs, butter, molasses, or beer? Does it matter?
Applesauce can be substituted for OIL in a recipe, not for eggs, molasses, buter, or beer
I'd probably start by replacing the beer with applesauce -- depending on how wet the batter is it could work as a 1:1 replacement, but you may need to add extra liquid if it's too dry. You can replace melted butter with applesauce as well, but I think you'll have less textural change by replacing the beer. If you go with the fat replacement, start by only substituting half (or less) with the applesauce.