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.