To reply to Shminux, I imagine it's because the sense of doom is some magical warning of the repercussions of Harry and Voldemort's magic colliding, which probably is not being set off by the probability of Harry's life being ended by a high-velocity piece of metal fired from a completely non-magical weapon.

To Alsadius, I think the gun is more deadly to Harry than an Avada Kedavra because of Harry's previous resistance to the spell. Remember when Quirrell previously suggested that they stage a "return of Voldemort scenario" where the pretend-Voldemort shot another killing curse at Harry who would block it with his Patronus 2.0? Harry's reaction was something along the lines of, "Nobody would believe that Voldemort would be so stupid to try that again." Apparently, Harry's ability to block it (as demonstrated in Azkaban) and previous resistance (as demonstrated the night his parents died) made Voldemort think the same thing.