Both abilities are nearly identical. But they have different names, give unnamed bonuses and because of that should stack. Or am I wrong?
Swashbuckler's initiative:
At 3rd level, while the swashbuckler has at least 1 panache point, she gains a +2 bonus on initiative checks. In addition, if she has the Quick Draw feat, her hands are free and unrestrained, and she has any single light or one-handed piercing melee weapon that isn't hidden, she can draw that weapon as part of the initiative check.
sleuth's initiative:
At 1st level, as long as the sleuth has at least 1 luck point, he gains a +2 bonus on initiative checks. Furthermore, if he has the Quick Draw feat, his hands are free and unrestrained, and the weapon is not hidden, he can draw a single light or one-handed melee weapon as part of the initiative check. (Unlike with swashbuckler deeds, this does not have to be a piercing weapon.) At 4th level, a sleuth gains the following deeds. These abilities replace swift alchemy.
I don't see any reason why they wouldn't stack. They are clearly different sources, even if they were both modeled after the same ability.
Yes they stack. It is an untyped bonus from two different sources.
I just wanted to be sure that it doesn't count as the same source because both are (nearly) identical. But I'll gladly take the +4 to initiative for my inspired blade/sleuth.