I will add that I could not use that carrot on a stick all the way thus depleting it and getting a fishing rod back.

I place a red carrot on a stick in my inventory and expected it to stay red while I got another one out.

I placed the carrot on a stick in my inventory and it replenished to half durability.

1. Get a carrot on a stick down to the point in which the durability bar is red
2. Click on it and put it in the inventory (not the quick bar)
0. If the first two don't work try putting the red durability bar carrot on a stick in the crafting menu, and add an undamaged carrot on a stick to the crafting menu, remove both and the red one will be half durability.