So, by syncing clock to an agreed-upon pulsar, both the Imperium and the Terries would have the same reference.
But would the Imperium base their second on a sub-multiple of 1,000,663,527,397 pulses while the Terries base theirs on a sub-multiple of 1,000,000,000,000 pulses from the same pulsar?
I've never read anything about a conversion factor for Imperial seconds or Solomani seconds. The joke about the Solomani second being larger goes here.
With all of the units in physics based upon seconds, if they're not the same, you'd be running conversions for every darn thing.
Which may or may not matter, depending how devoted your world was to the Solomani Cause. "Only Impie-loving backsliders use metric. We use something pure, ancient, and undeniably Solomani - feet! Pounds! Inches! The Imperial System!!"