The main bottlenecks with this possible development is solving the problems to build viable sublight interstellar spacecraft and the capacity to produce enough antimatter to power them. Unfortunately the two problems are interlinked.
In 2006, Gerald Smith estimated $250 million could produce 10 milligrams of positrons.
in 1999, NASA gave a figure of $62.5 trillion per gram of antihydrogen
There are possible ways of bringing down the cost of antimatter, but unless a radical new technology can be devised to do this cheaply antimatter will put a brake on the energy and cost budgets for interstellar space travel.
it might be possible to use magnetic scoops to collect the antimatter that occurs naturally in the Van Allen belt of the Earth, and ultimately, the belts of gas giants, like Jupiter, hopefully at a lower cost per gram.
It is plausible to assume the biotech and AI's proposed by the mid-23rd century. Even making both more highly advanced than proposed.
To reach stars up to sixty light years, and within reasonable timeframes for a thriving interstellar economy, suggests 23rd century starships travel close to lightspeed. This will take vast amounts of antimatter to propel them. However, if the starships travel at lower velocities, less antimatter is needed to fuel them. The trouble with this is that round trip times soon push out to longer timescales.
For example, if the starships travel 0.5 c, as a matter of antimatter economy, then a trip to a star sixty light years gives a round trip time of 240 years. A starship launched in 2247 will return to its home base in 2487 plus however long it remained at its destination. This might apply to the limits of interstellar exploration rather than to an interstellar economy. Even journeys to nearby stars at 0.5 c, will have round trip times of decades. Not impossible or too prohibitive, but if it's factored into your future scenario then you can know what to expect.
With the most optimistic projections for increased, and cheaper too, antimatter production and the rapid development of interstellar space technology and vehicles by 2247 it is plausible we will see the beginnings of an era of interstellar exploration and an attendant interstellar economy.
However, it is plausible that the proposed biotech and AI's will exist by 2247, and might even more advanced than the OP suggests. This can be tweaked to meet the requirements of the future scenario.
Despite any arguments to the contrary, it's still your world and you're welcome to design whichever you want. These are only my projections. Besides knowing what other people's objections are can be basis for finding ways of overcoming them and building something that will do want you and be more plausible.
It is worth considering there might be a wild card factor that has greatly accelerated the drive to go into interstellar space. For example, the discovery of aliens with their own thriving interstellar economy that has expanded out to sixty to eighty light years, but this is located at a safe distance of several thousand light years. Humans will want to carve out their own sphere of influence in the galaxy before other sapients take it away from us.