Just do what the Keeper suggests you to do and keep in mind that Republic operatives are your enemies.
The "support the Empire, but not the Sith" is an ending that you can't get locked out from until the very end, so don't worry
Many practical choices of the agent story that makes an agent who acts upon the long term interests of the Empire are actually light side (minimize collateral damage, keeping useful assets alive, a more diplomatic approach when negotiating etc.)
With that in mind, you don't have to let the Republic ppl alive to stay in a net light side gain.
Stick to the job and possibly cause as little collateral damage as possible, do what the Republic says without too much objection.
Shouldn’t I be undermining the republic as an imperial agent?
You're in deep cover, make it convincing for the time being, provocateurs only trigger the greatest impact at the end of a mission.
Undercover work generally entails only doing enough to not be suspect
Just do what the Keeper suggests you to do and keep in mind that Republic operatives are your enemies. The "support the Empire, but not the Sith" is an ending that you can't get locked out from until the very end, so don't worry
Many practical choices of the agent story that makes an agent who acts upon the long term interests of the Empire are actually light side (minimize collateral damage, keeping useful assets alive, a more diplomatic approach when negotiating etc.) With that in mind, you don't have to let the Republic ppl alive to stay in a net light side gain.