Nets do not have any special factor that removes this possibility.
The reason the Lucky feat works to replace disadvantage with "super-advantage" is because:
You choose which of the d20s is used for the attack roll, ability check, or saving throw.
This means you are choosing between which of the three d20s (two from disadvantage and one from Lucky) even though usually you would take the lowest of the two from disadvantage. Giving you the ability to choose is what causes this to negate the disadvantage.
Nets are in an interesting state since they are almost always at disadvantage as you can either attack within 5 feet, or attack at long range. However, the Lucky feat still lets you choose which of the three d20s is used.