A Birch planet cannot be natural, it is a megastructure that compares to a giant Dyson Sphere. Suppose your civilization would be able to create a Dyson Sphere the size of Neptune's orbit, it needs galactic amounts of iron to construct it. Hundreds (thousands?) of stars and gas giants would have to be dismantled to build it. But.. there are more issues.. I have to shoot down the whole idea, a Birch planet is imho an implausible construct.
Frame challenge
No sunshine and a lot of dangerous impacts
In the usual Dyson sphere, you live on the inside. That works, because there is a sun inside the construct and a Dyson sphere will be rotating to generate "gravity" directed outward. A Birch planet, or Birch Dyson sphere can't possibly rotate fast enough to compensate for the inward gravitational force. As a result, you'll have to live on the outside, where you got your 1G gravity (Neptune orbit). Your construct will receive no sunshine - or too much sunshine - and it will be easily perforated with accretion disk material falling into the black hole, at near light speed.
Time dilation
An object approaching the event horizon would appear to be slowing
down as it approaches the black hole instead of speeding up as we
would expect. This is because of time dilation, which slows the
passing of time for an object moving close to the speed of light with
respect to an outside observer. The object eventually would slow down
until it stops moving when it reaches the event horizon
http://ffden-2.phys.uaf.edu/webproj/211_fall_2014/Chris_Bon/chris_bon/blackholes.html
Yes, your computers will run faster, but that only counts for the outside observer. The inhabitants of your Birch planet won't notice that. Also, you'll have a problem with communication. For an outside observer, perceived time on your Birch planet will be slowed down. As a result, your radio waves will have a frequency well beyond the receiver capabilities of the observer. The closer your Dyson sphere will be to the event horizon, the worse that issue will become.
Escape will be difficult
Your civilization has built a mega structure around a black hole, they will want to do space travel. With 1G gravity, you'd assume that could be arranged. But you'll have to take into account this 1G gravity will not diminish outward, like it happens when you leave planet Earth. The 1G gravity will extend outward for many AE's, your space vessel would need to maintain escape velocity for days on end, to get away. You'll need a special drive or zillions of liters of fuel, else you can't possibly escape the place. Time dilation will spoil your space travel as well: the nearer your Birch surface is to the event horizon, the larger the time difference your space ship will perceive when it looks back. And the longer it will take your ship to return. Space travel will take forever, for anyone on your Birch planet.