Kanderas, I'm going to add a second answer here, because I didn't directly address your question; rather, I pointed out why that scenario couldn't be reconciled with what we see in our sky.
But what if we ignore that? How would things be different in your scenario, and what can we do to minimize those differences?
The main problem I see, if the scale on your diagram is correct, is that the day/night temperature difference will be much greater, with the sun advancing and retreating by a large degree every day. Such a large temperature difference isn't compatible with the stable, mild conditions necessary for complex life to evolve, nor in fact with retaining an Earth-like atmosphere. If the uninhabited side is baked by unending daylight, it will soon be heated beyond the boiling point of water, and so lifeless as the moon. Worse, on the very hottest parts, it might get so hot that even water vapor would be disassociated into hydrogen and oxygen. If this happens, the hydrogen would escape into space, and slowly over the eons the planet would become lifeless as Mars.
Contrariwise, if the "central fire" is just a massive object but not a sun radiating warmth, the dark side would freeze so hard that the atmosphere would freeze there, and it wouldn't be long until the entire planet's atmosphere (and hydrosphere -- the water, lakes, rivers, and oceans) would vanish away.
In either case, either a terrifically hot or extremely cold "dark side" would create eternal hurricane winds flowing into the cold side, or away from the hot side, all along the terminator and probably thousands of miles into the inhabited side. If you're getting the idea that in this case, "inhabited" means "not very inhabitable"... you're right. This isn't a scenario for a habitable planet.
But there are other assumptions we can make with much more temperate outcomes, so let's make them. Let's assume the "central fire" is a red dwarf star, emitting enough light (and infrared radiation) to keep the hidden side of our planet pleasantly warm, at a stable temperature that just happens to be the same as the average temperature of the inhabited side. Let us furthermore assume the Earth is in close orbit around this red dwarf, fitting your tidally locked scenario.
Now, what we need to do is greatly expand the distance between the Earth's orbit around that "central fire" and the Sun. This has many benefits, not the least of which it greatly minimizes the objection I made in my first response. If the Sun's orbit is quite distant, relative to the Earth's orbit around the red dwarf, then the growing and shrinking as the Earth approaches and retreats from the sun will be unnoticeable, or at worst barely noticeable. Since in our universe, the Sun is some 93 million miles away, we have lots of room to do this and keep it realistic.
In this scenario, we do lose our moon. A tight orbit around a red dwarf would be much too close for a stable large moon at a large distance, like Luna. A smaller one would be lost fairly quickly (on the geologic time scales) because of tidal dragging by the red dwarf.
Of course, we can expand the Earth's orbit around the red dwarf and keep Luna... and also make more room to squeeze in Counter-Earth in an even closer orbit. But then an observer on Earth would start noticing that the sun grows and then shrinks as the day passes.
If there is enough difference in the apparent size of the Sun to be noticeable, then there will be enough difference to cause a greater day/night variation. Even if the "dark side" of Earth doesn't freeze or roast, there still will be enough variation in day/night temperature to significantly affect evolution. Of course you can arbitrarily choose any evolutionary history you wish, since it's your world. But realistically, we'd tend to see much less variety in land animals. I'd suggest no land animals more complex than insects, and less variety in plants, too. Keep in mind that trees and grass are products of fairly recent evolution, so aren't likely to exist. Unless you plan to make the "people" in your story sea-dwellers, there's not going to be a lot of story-telling potential in this scenario.
The trick here will be trying to choose a "Goldilocks" distance for the Earth to orbit the red dwarf; far enough away to allow a stable Luna orbit, but close enough to minimize the variation in the Sun's apparent size. This would be far easier if you stipulate that Luna's orbit is artificial... that it's the product of some long-vanished alien race which buried a reactionless drive deep in the moon to keep it in that orbit.
Now, about using a black hole as the "central fire": That's a very bad idea, from the standpoint of scientific realism. Black holes have accretion disks pulling matter into them, and they emit very high energy radiation. (Even the thin interplanetary hydrogen and dust will be sufficient to create a weak but permanent accretion disk around the black hole.) They also have searchlight-like jets of high radiation and plasma emitted from their poles. Thankfully that wouldn't directly impact Earth, because Earth's orbit should be roughly perpendicular to those jets. But even the secondary radiation from the black hole and its jets would do really nasty things to the Earth's atmosphere, and remember that winds will flow from the inhabited side to the "dark side". Water will flow around the world, too.
So, my advice is to not try to use a black hole as the "central fire". It's just too nasty.
I hope you realize that in any variation, Counter-Earth is significantly closer to the "Central Fire", and thus has a significantly hotter surface. So any assumptions we use to make Earth relatively close to Earth-as-we-know-it will automatically rule out an Earth-like Counter Earth.
Seems a pity to have a Counter-Earth and have it be as lifeless as Venus or Mercury. Frankly, if it was me, I'd make this a fantasy world rather than a scientifically realistic one. Then you could have Luna without worrying about its unstable orbit, and Counter-Earth might be the home of ifrits and fire demons! Obviously the inhabitants of that Earth would find a different name for the other planet; it's hardly a "Counter-Earth"!
