Does Lure counter Luck of the Sea in any way?
According to Minecrafts Gamepedia Enchanting page, Lure only increases how often the fishing rod will catch something:
Increases rate of fish biting your hook
Decreases wait time until a catch by 5 seconds per level. At Level VIII, the fish-catching particle effects start almost instantly. At Level IX, you are not able to catch anything.
Meanwhile Luck of the Sea increases chances of treasure:
Increases luck while fishing
With the default loot tables, this lowers chance of "junk" catches and increases chance of "treasure" by about 2 percentage points, and decreases the chance of fish by about 0.15 percentage points.
So to answer the question, I really see no reason as to why Lure would counter the effects of Luck of the Sea. I think you are just really unlucky.
Note: According to a table on an almost duplicate post on Reddit, your chances of catching treasure are still at a low 11.34% (though much better than the standard 5%).
Your generic.luck, which is what determines how good the loot you get from fishing is, is only changed by the Luck of the Sea enchantment; nothing else in vanilla survival has any effect on it (and fishing is the only thing it affects). Lure does not affect the quality of loot you get, just how often you get a bite.
Similarly, Luck of the Sea does not effect how often you get a bite. There are however other factors that determine how long you have to wait:
- If your fishing bobber is not exposed directly to sun/moonlight, the time taken to catch a fish will double
- If you fish in the rain, the time taken to catch a fish will be 20% less
Keep in mind that fishing up a mending book is very rare.
With Luck of the Sea III, the chances of getting junk/treasure/fish are roughly:
- Junk: 4.1%
- Treasure: 11.3%
- Fish: 82.5%
If Treasure is chosen, your chance of getting an enchanted book is then 16.7%.
With some experimentation I found the chance of an enchanted book, enchanted with the same parameters as fishing enchants books with, having mending is 3.7%.
This all means overall that your chance of getting a mending book is 0.113×0.167×0.037=0.000698, or 0.07%.
This means that you will have to catch, on average, roughly 1,400 things before catching a mending book.