A creature becomes an attacking creature right after attackers are declared, and therefore can be the target of Impeccable Timing once that happens (C.R. 508.1k). But at that point, before any ability resolves (including Hooded Brawler's delayed triggered ability that would give it +2/+2 until end of turn), players have priority to cast spells, including Impeccable Timing (C.R. 508.2, 116.1a, 116.3d, 116.4), and such a spell will go on the top of the stack (C.R. 405.2) and get to resolve before that ability (C.R. 116.4).
Note however, that Impeccable Timing deals only damage, and that it's not damage, by itself, that destroys (not: kills) a creature, but rather a state-based action could do so (C.R. 119.5, 704.5g-h).
EDIT (May 4): One rule was renumbered with Amonkhet.
EDIT (May 10, 2018): Add rule citation; correctness edit.
Exert is an ability that goes on the stack, so your spell would resolve first, killing it before it got the bonus.
Note however, that Impeccable Timing deals only damage, and that it's not damage, by itself, that destroys (not: kills) a creature, but rather a state-based action could do so (C.R. 119.5, 704.5g-h).
EDIT (May 4): One rule was renumbered with Amonkhet.
EDIT (May 10, 2018): Add rule citation; correctness edit.