No, Lugia is not shiny-locked and, as you said, the probability of getting a shiny is 1/4096, or 1/2048 with the Shiny Charm. With soft resetting, the shiny rate doesn't change; each Pokémon you encounter has the exact same chance as the last.
But even though these are the odds, you're not guaranteed a shiny if you SR that many times. Your question suggests that you might not understand how probability works. As a simple example, let's say that you have two Pokémon with a 50/50 gender ratio, and you're looking for a female. There are four possible configurations for what their genders could be:
As you can see, only three of these outcomes contains a female, so even though there is a 1/2 chance of getting one, in a group of two, there is only a 75% chance that one of them will be female. The same logic goes for shiny Pokémon. That's why some hunts yield a shiny in just over a hundred encounters while others can last well over 10,000 (I've even seen some stretch into the 200,000 range).