At 2.3% energy use would double every 30 years. So in 350 years it will have doubled 12 times, or be about 4000 times present day usage.
2015 usage was 49 million TWh (terawatt hours) all energy uses.
That works out to 5,000 TW. (8800 hours per year)
Intercepted sunlight equals a disk with radius 6300 km so it has an area of 125 million square km or 125 * 10^12 square meters. Sunlight that makes it to the bottom of the atmosphere is about 1000 W/m2. So the earth receives 125,000 TW on a continuous basis.
So we have a surplus of about 4%. This is equivalent to about 40 W/m2.
If the earth were a simple black body the temp would have to increase by about 1% of it's kevin temp to compensate. (It's a 4th power law and to first order 1.01 to the 4th is 1.04) Earth right now is about 300K so a 1% increase is 3 C.
That would be tolerable.
But right now we have a surplus from the extra greenhouse effect of somewhere between 1.5 and 2 W/m2 depending on who's model you use, and that's giving us somewhere between 1 and 2 C warming. On that basis 40 W/m2 implies a 20 C average increase which would only leave the poles habitable.
The model would break down before that. As temperatures increase evaporation climbs very fast (7%/C) and so the fraction of the earth covered in cloud increases. This increases the albedo.
So you have a wide latitude to pick your scenario. I would suggest a geometric mean between 2 and 20 or about 6-8 C
The following does not address your question, but questions your world:
Cheap fusion power means that removing the CO2 from the atmosphere becomes practical. Without the greenhouse effect earth would be some 18 C colder than now. So by sequestering carbon, you could control the global climate. (Capture CO2, reduce to carbon, make into bricks -- synthetic coal -- and store underwater so it doesn't catch fire.) This could become a plot element.
Cheap energy makes an awful lot of other technologies possible too. You have to address your entire world and the history to see what the impact of cheap energy would be on that problem. E.g. land reclamation is easier with cheap energy to move stuff around. The muck off the coast of the world's large rivers will fertilizer a lot of earth. Producing synthetic fertilizer is dirt cheap.
You also have to explain what has happened generally in technology. The changes in the next 350 years should be as strong as in the last 350 years. Where were we in 1650?
Our understanding of ecology is growing, but no where near being a predictive science. Given enough sensors and enough computing power, howerver it should be. Why is your complex natural/human ecology not amenable to this.
Why is access to open space limited? Even if there is a lot more desert, deserts are tourist attractions. See the travel literature for Utah; for the Gobi desert. Are our present cities gone? Why? (Many because they are underwater -- all the ice melting raises oceans by 200 feet. This also means you have a fact check for each existing city name you use. Denver is safe, as is Chicago. Seattle, Washington DC etc are not.) Would not existing cities become tourist destinations? The more equitorial destinations for adventurers in heat suits, and super air conditioned vehicles.
Our understanding of genetics is growing fast. Why have the deserts won? Combine genes from things that live in hot springs, and make a plant that can work at 50 C.
Maybe the deserts didn't win. People use virtual holidays because they are cheap, or there are never lines, or they can be more exciting, or the sex is better, or that you never lose your luggage, and the Danish deserts are non-fattening.
Cheap energy makes manufacture and remanufacture much more attractive. Add to it robotics and AI, and you have goods so cheap that you aren't paying for goods you are paying for fashion. (This is already true. In constant dollars off brand jeans are a fraction of the price they were when I was a kid.) You need to explain (at least to your self in your background) why this didn't happen, or what form cheap goods has taken.
If owning something no longer has prestige, how do people show off? In 1650 being fat, or at least plump was a mark of wealth -- you could afford to eat. You traveled by coach, with a matched pair of horses. One or more footmen. Dalmations running on either side. You had lots of clothes, because even the wealthy didn't have central heating. Now we struggle to be thin; if we are successful, we flaunt it with next to nothing running shorts. The coach has been replace with a Porche. And being fashionable is expensive, and for many is how we show off.
In general with increased food security and education, population growth rates decline. What is the history of your world that you have a high population density? In many first world countries, population isn't replacing itself. We have 1.9 births per woman. Replacement requires between 2.1 and 2.2.
The high tech distopia is a staple of Hollywood, but needs to be crafted with care. A high tech society requires lots of technicians to create and maintain, and at least some training of the masses to use. A high tech world of uneducated people is only possible with AI+robotics doing the support work. You can have a high tech society with educated people living in miserable conditions. E.g. zero privacy, massive crowding, short food or just yeast mash for food, but if this is an extrapolation of our present world, you need to fill in why. (Note you don't have to do this in the first book. Veiled hints and obscure references that don't make a lot of sense until the second time people read your decology are allowed. But YOU have to figure them out ahead of time to make your story consistent.)
