The sheer closest I can fathom for an empire limited by the speed of light is <PERSON> from A Deepness in the Sky. And I swear I've writtenabout them on WB (or atleast on SE somewhere) before, but I can't find it. A few other people have with regards to a few topics.
The Qeng Ho weren't an empire as such, but a loose collection of disparate groups that all held the same ideals and goals; essentially a societal norm spread across light-centuries.
These groups would fly between known civilized worlds and trade in whatever might be of value. Sometimes it would be technology, or it might be goods, or it could be information. Occasionally they would arrive and find the planet having bombed itself back to the stone age and the Qeng Ho would aid in rebuilding a space faring society because they knew that in 100 or 200 years another Qeng Ho ship (them or someone else) would be by again and the planet would have valuable trade once more.
The backbone of what made them functional was--effectively--a galactic radio station. Every Qeng Ho ship would broadcast an automated signal that would contain the blueprints necessary to go from radio ("are you recieving? Build this...") to intra-solar space travel. 302
As well as things like language and the society of the Qeng Ho themselves, so that when the traders showed up, they could parley with minimal effort.
Underlying that was a private Qeng Ho encrypted channel that allowed groups to talk with each other. Not quickly, but still faster than their ships could travel. They'd use this channel to broadcast locations of new civilized worlds, premium technical advancements (the <PERSON> kept the best stuff to themselves), and so forth. They knew that any given message may never be received, much less generate a reply that they would hear in their lifetimes (even long as they were due to relativistic time dilation and life prolonging technologies).
But I wouldn't call them a K3 civilization, necessarily... Just the closest thing to "a galaxy-sized society limited by the speed of light" that I am aware of. And I'm not 100% sure that this would truly work, but it is plausible enough for a novel. 209