Currently, we start the quest chain by getting to Rank 7 in the record book and using Xur's tag to open the middle large present in the Tower. After the Dawning event ends, the presents will likely disappear. How will we get the quest afterwards? Does anybody know at this point, or is it something Bungie hasn't revealed? I mean, it'd really suck if they were unobtainable after the event. I'm also fairly sure the fanbase would go ballistic if that were revealed to be the case.
You are waaay too far ahead on this one... worry not and get your gun quest within these 3 weeks. I'm sure the quest will expire when the event's over & they'll just be thrown into the current exotics RNG pool.