Actually, Chiss Imperial agents get a lot of dialogue where they're specifically called out to be Chiss, and you even have a little background story about where you came from, and talk about being from The Chiss Ascendency, Csilla, etc..
But that's ONLY Chiss agents, you won't get that in any other class as Chiss.
But to the OP, no, unfortunately the republic won't notice that you're playing a "Imperial" race.
I still remember a patch note about some IA storyline NPC "no longer addressing all alien players as Chiss", or something like that. Watching my girlfriend play her Chiss agent and seeing how specific it got made me wonder what dialogues come up for, say, a Rattataki, but I'm too lazy to reroll mine to find out.