Junk is any of the following items:
Bowl, Fishing Rod (Highly Damaged), Leather, Leather Boots (Highly Damaged), Rotten Flesh, Stick, String, Water Bottle, Bone, Ink Sac & Tripwire Hook
Treasure is:
Bow, Enchanted Book, Fishing Rod, Nametag, Saddle, Lily Pad, Fishing Rod (Enchanted)
With an unenchanted Rod, your odds are: 85% Fish, 10% Junk, 5% Treasure. With Luck of the sea I, it goes to 86.5%, 7.5%, 6%. Luck of the Sea II: 88%, 5% 7%, and Luck of the Sea III: 89.5%, 2.5% 8%.
If you get treasure, your odds of getting a name tag are 16.7% (1/6), so with Luck of the Sea II you have odds of 1.2% of getting a Nametag each time.
The only way to increase this would be to get Luck of the Sea III, which increases the odds to a whopping 1.3%.
You are using the right fishing rod. There is one change in 1.9 that made the reddit post obsolete:
Minecraft 1.9 Changelog:
15w43c Lure no longer affects loot.
It means that the reduction of 1% per level on treasure is no longer in effect. So the faster you fish, the more treasure you get.
The best fishing rod for catching treasure then has
Luck of the Sea III
Lure III
Adding mending and unbreaking while a good choice to save experience points, are irrelevant to the chances of treasure.
Best Answer
As of 1.9, the Lure enchantment does not modify chances of fishing and only increases fishing speed. You can safely combine Luck of the Sea and Lure to have full benefits of both.