Turkish citizens eligible

I have to admit that I am really puzzled with your statement that Turkish citizens are not eligible for individual tourist visas. My knowledge was that they are eligible for any kind of visa, but they are required to make the application in their home country - Türkiye. If that is the case and I hope you have checked well with the embassy/consulate general and not from different unreliable sources, then you can proceed with your plan to buy an additional return ticket Beijing - Seoul - Beijing, but it should be definitely done from the different airport as you have already suggested.

By combining PEK and PKX airports, the airport authorities will simply have to issue a temporary entry permit because you have to formally enter China in order to accomplish your flight itinerary. I also advise you to try to maximize the amount of time you will spend in Beijing i.e. that your scheduled layover becomes as close as possible to 24 hours.

So, when you land in Beijing, the first thing you will do is to approach 24/144 hour visa free transit counter where you will apply for a temporary entry permit by presenting your flight tickets and hotel booking if your layover is during the night hours. I strongly advise you to print out your flight tickets and your hotel bookings. After a short check, you will proceed to the immigration and then to the city to enjoy the time on your own. The same thing is valid for your second arrival in Beijing.

What to tell you, but to have a pleasant, although in many aspects very specific journey.