So we are in need of some urgent help trying to figure out what category this beer we brewed is. It came out first and foremost delicious, but also super smooth, malty, more oak chippy than a bourbon taste (but the bourbon is coming on more as it ages). Here's the recipe:
The first and most important thing to consider is "what does the beer taste like?" The judge doesn't see the recipe.
Just because the beer touches wood does not necessarily make "wood aged" the best fit category for that particular beer. In any of the speciality category the special ingredient / feature needs to be front and center.
That said, based on the recipe I'd agree with the earlier poster. Old Ale might make sense. It is a category with a fairly broad range of possible interpretation, and a hint of oak would not be out of place.