My take on it.
Just about every CT source is written from the Third Imperium's viewpoint, so they often don't tell the whole or even a true story - there are secrets such as the Aslan's minor status, the Zhodani not being bad guys, that sort of thing. The Solomani are not the fascist/communist/totalitarian state that the Imperium want to portray them as.
They are a confederation, some Solomani polities are democracies, some ore corporate empires etc.
There is even the myth of a long night. There was no long night from the Old Earth Union point of view - the First Imperium was conquered, the Terrans took over for a while but the far away Imperium fell - effect on Earth and the pocket empires springing up around Earth, next to nil.
The Third Imperium invented the concept of a long night for the Sylean region of space because that is what they suffered. While trade within the region of the first Imperium was now just about dead, the far off Solomani pocket Empires continued to expand - they encountered the Aslan and began border wars with them - not exactly the actions of a defunct polity.
Was there technological stagnation within the Solomani area during this so called 'long night' period? Probably not too much actually. By the Time the Sylean Federation - sorry - Third Imperium recontacted the Solomani region there were probably many TL13 or even 14 worlds within Solomani space.
The Third Imperium has two things to offer, fusion+ and a much larger economy to trade with and partner with. That the Third Imperium was heavily 'solomani fanboy' (and of solomani ancestry) probably helped a lot, the Imperium wanted the Solomani to join, which they agreed to.
We know from Agent of the Imperium that you should take the overall TL of an empire with a pinch of salt. The supposedly TL13 Imperium was using jump 5 engines in scout cruisers long before the empire as a whole was elevated to a TL14 base, and similar jump 6 ships would appear by the 700s, three centuries before TL15 became Imperial standard.
The Imperium has one other ace - they have access to the secret Vilani database repositories of technologies and cultures encountered by the Vilani during their five millennia in space - the Vilani locked these secrets away, frightened of the paradigm shift TL12 would bring (I have an essay on that

) and deliberately stagnated their TL to 11.
The Syleans had no problem adopting TL12, quickly elevating themselves to TL13 and then experimenting with TL14 all within a few centuries of the founding of the Imperium.
Those secrets allowed Imperial r&d to perfect the meson screen, nuclear damper technology, access higher jump dimensions, but the Solomani didn't have the secret Vilani repositories to call upon for the rapid development of the 'black ops' Imperium for want of a better term.
The Third Imperium did still shy away from intelligent machines - artificial sentience may be TL16/17 but you can emulate/run a human personality from TL13, build robots and computers that can learn from TL11/12 - there are examples in AotI of TL12+ world computer networks achieving sentience and the world having to be scrubbed as a result.
Meanwhile the Solomani, now within the Imperium, consolidate their area as TL13, are slowly building to TL14 all through original research. Rather than rely on machines the Solomani go in for genetic engineering and their much greater average world population - people do the jobs instead of machines.
The Solomani Rim War was the major reason for the Third Imperium to push their industrial base to TL15 - the early war had a TL13/14 Solomani fleet facing a TL14 Imperium. The Imperium had the r&d to roll out TL15, the Solomani didn't.
Post Rim War the Solomani consolidate to T14 and begin the climb to TL15 on capable worlds, but they are a way off having a TL15 base.