Okay, so non-enchanted bows are affected too. But this time I found the real problem.
You hear the "break" sound, but it stays in your hand and your hot bar until you right-click for any reason.
So my guess is that when I ran into the bug earlier, I may not have right-clicked the bow after it "reset" itself, I must have selected a different item in my hot bar at the time. Thus, the nonexistent bow remained there until I tried to use it on the enchantment table. Yet it was able to work on the anvil. (Though I probably didn't have it selected in the hot bar at the time.)
Bottom line, the bow becomes a ghost after it breaks, enchanted or not. It stays in the hot bar and in your hand until you attempt to use it. You can mod it on an anvil granted it's not selected in your hot bar at the time.
I changed the thread title to to match the real problem.
So I tried to replicate the bug and I got a result very similar to what I had before.
The bow reset itself after using up the durability. Yet this time, it disappeared when I right-clicked the anvil (to add the Power V enchantment).
So yeah, I think the problem is a delayed reaction when the bow breaks.
I'll come back after I try to see if the same problem occurs with a non-enchanted bow.
Nowhere to be found. I added it to the box under the "Enchant" and boom, disappeared, just like that.
I added more details in the description. Hopefully that helps.
Confirmed as of 14w02c on single player creative mode.