Basically in order to stop FR or to stop CR becoming FR you need to kill both ; not just because they are different beings but also because they are on different time lines which can be reset as long as one being exists. If FR is killed and CR isn't CR can potentially become like FR if certain criteria are met , but if CR does not meet the set criteria he may follow the other path , the alternative future in my opinion will be a mix and match of the two futures put together.
Just think a bit like Dr Who... Time is always changing ; one past event can change the course of time throughout the entire universe forever. If CR is to die now:
Sting will not be dead in future, FR will not exist (on one time line), the current events would never have happened due to the fact CR does not exist eliminating the possibility of FR from his timeline , he will still exist as a completely different being in an alternative timeline not linked to the events ; however this then causes a problem... It means that Eclipse would never be used ; it still may be the last hope to save both timelines from the destruction which is about to occur within the following seven years.
Which is why i think FR will be killed , CR will be spared , but his shadow (who seems to possess Rogue and probably causes CR to become mad like FR) can be killed. If this is the case one time line is now no longer possible. Therefore leaving the unknown alternative and FL's time line as possibilites.
If FL's timeline occurs the dragonslayers will need some dragon friends to find their parents ; enhancing their strength making them strong enough to kill dragons easily. And on top of that bringing them closer to being capable of defeating Acnologia (one of the series primary goals).
Here are two possible theories, hell they could actually be true in real life but aren't falsifiable so they can't be accepted as having credibility.
time travel theory A: There is only two paths, the one the traveling person came from and the one he arrived at. The only way for him to go away is to change the current world enough so that he does not feel the need to travel back in time. FR being the time traveler. I'm just guessing that in FR's time there were 0 dragons, and the 10000 dragons was a myth. because FL basically had no effect on anything I guess we can ignore her (though that may change because of her diary that CL found)
FR came back to our point in time, got 7 out, if that was enough to destroy the world then on its own then future FR wouldn't need to go back in time and cease to exist in our world. To me that means the 7 won't destroy the world right now something else has to happen to make future FR want to come back to try and control the world. The only way to stop FR from coming back in time is our world be destroyed to his liking or CR dies before the date he decided to go back in time. Killing FR will do nothing to save the world because if CR is still alive in 798 he will travel back to 791 to try and gain control of the world.
edit: This idea is the one that the characters believe and Mashi is trying to push on us but it isn't well thought out.
time travel theory B: There is an infinite number of universes that are changeable but only changeable in one universe. The one you came from stays the same. Basically think of that as every single instant after the big bang a new world is created. FR and FL came from two that started 7 years earlier than the one we are in now. In our version of the world we have two people, FR and FL, arrive to tell us of the past of their universe and try to change our future. Whatever happens happens blah blah blah. If FR and FL returned to their universe it would remain unchanged. We were just lucky enough to be picked and our universe changed, The one FL came from wasn't so lucky, and the one FR came from wasn't so lucky (for him). It was just a coincidence that two future people showed up in the same universe. Maybe the universe born a second after us had 3 people come from the future. this idea is the one that says there is no free will everything that happens happens and couldn't of happened any other way. This is what I believe and tried to describe earlier.
I call this the Trunks from DragonBallZ theory.
Mashi's Theory: By having more than two futures I would say he subscribes to B, but the characters think that A is true. For example, if you change CR and CL's future for FR what happened to FL's, nothing?, Her diary may help CL but it isnt going to help FL's Levy
It still makes zero sense that FR knew about the gate considering CR was no where near it.
Whatever, it's just anime, my head hurts to much now. This is why time travel is stupid. So many different TV shows, movies, scientists, normal people, anime, manga etc have a theory about time travel. Talking tacos that crap ice cream is more plausible in a Fantasy story.