I don't think Goryeo will not follow this scenario (my TL) because best scenario around 1264 is Kubilai asking assistance from them and in return give Liaodong.
1. Bureaucrats wanted peace, heck they even submitted too Mongolia in 1259. So this is gift from Heaven.
2. For Military Government it is best they hope. You will raise your status by helping Kubilai and If they can gain Liaodong they can legitimate their control, hence support of population. At worst they will have more time to sort out their internal matter, and with the Kubilai money they could fix their military.
Many members of the aristocracy wanted peace after years of chaos, but not all did, as the military regime's influence remained strong. In fact, when the Mongols requested closer relations in 1264, the military and some nobles began to form an Anti-Mongol faction once again, mostly due to the fear of supplying troops for a potential expedition to Japan. In other words, if the government was wary about supporting a joint expedition to Japan, in which Goryeo would potentially gain territory, I honestly don't see why they would supply any troops to one side in a civil war with an extremely uncertain outcome. The generals would almost certainly not risk potentially severe repercussions in order to gain so much territory, and the fact that Kublai did not request for outside support IOTL suggests that he wanted to handle it on his own.
As for Kubilai it is not make sense. But you can argue that, he wasn't serious about his promise (it is not uncommon among nomadic tribes). He wanted to deal with Ariq Boke, then after that he can invade Goryeo anyway. So it is not impossible.
Kublai wouldn't want to make a promise that would make Goryeo stronger, even on pretenses, especially if he's dealing with a state that continuously resisted for almost 30 years at the time and caused the invasion of the Song to be delayed. The fact that Goryeo's northern regions were initially put under Mongol control only strengthens this assumption, and if he attempted to invade the peninsula once more, he would almost certainly face another civil war by other rulers and princes who would attempt to take advantage of the turmoil. His invasion of the Song would also be severely delayed, allowing them to counterattack more efficiently, not to mention that even the pretenses of an attack on Dadu would severely undercut Kublai's legitimacy as the ruler. There's a very good reason why out of all of the states that surrendered within Eurasia, the Mongols decided to only let Goryeo retain its existence, and Kublai would not attempt to make such a suicidal move in order to break the fragile balance.
In addition, your map suggested that Goryeo/Joseon held Liaodong for centuries, based on the expansion within Manchuria, which technically contradicts your suggestion that it can be taken back by Kublai once the initial succession crisis is over. If Kublai invades Goryeo, which in itself is close to impossible, the latter would almost certainly suffer large losses in territory, and no amount of pleading would convince the Mongols to return it until the Yuan collapses. In other words, Goryeo would have a hard time retaining territory
south of the Yalu River if it loses Liaodong around 1265-75, let alone even
attempting to expand further north.
And after Ariq Boke and Kubilai stalemate (under pressure of Royal Family Princes and Mongolian Nobles), Kubilai will be very interested to give Liaodong to Goryeo (now most certainly tribute state) rather than give it to Ariq Boke.
What wonders me is, how Gorey will capitalize this new base. They sure will be busy for several century to assimilate Liaodong rather than project power unless it threatens your national interest. If Japanese conquest of Korean Peninsula occurs as OTL, Korea will be introvert well untill when some of colonial power open them. After that everything will be really tricky.
Even if he attempts to give Liaodong to Goryeo, which in itself is close to impossible, he will probably only hand over the coastal regions, or a very small amount, in order to keep the state in check. Just taking that region isn't particularly hard for Goryeo to maintain, as it managed to double its territory within 50-70 years or so from 1290 to 1360, and retained most of it even during the turmoil within China, and the incursions from the Jurchen afterward. The issue is why Goryeo would accept such a dangerous offer, and why Kublai would want to give the territory, which is unreasonable in itself, then attempt to make a suicidal move which may cost him the throne altogether.