In "Mr. Sunshine", almost half of Korean/Joseon ministers were pro-Japanese and even Japanese ministers were allowed in the royal court. How accurate is this depiction of turn-of-the-century Joseon dynasty politics?

Sort of true.

Kingdom/Empire of Joseon was basically medieval kingdom thrown into Imperialistic death match environment, and they had China, Russia, Japan, USA trying to rip Korea into pieces.

Daewon-Gun, Father of Emperor Gojong was great political reformer, but he chose staying in medieval way and he failed. Also, he was economic disaster to Joseon.

So Emperor Gojong had to deal with all of foreign invaders with almost no modernization , knowledge or budget and 4 countries who seeks to colonize Joseon. He tried to deal with this desperate situation with neutral policy. He held balance of power in Joseon by distributing powers/interests to each countries and their Korean supporters. I think he tried to make Joseon like Swiss.

But Japan was most vicious and fast-moving player in this game and they were just too close to stop. They broke into palace and butchered Empress Myeongseong(Queen Min) who was influential pro-Russian politician herself (Eulmi Incident, 1895), and won in costly Russo-Japanese war(1904-1905).

In such situation, Joseon had many politicians who were vile collaborators or who supported alliance policies, and pro-Japanese was biggest, also, most detested party. Of course, they've got Japenese Army behind their back. Japanese Army was maneuvering freely from Donghak rebellion(1894) and their officers were in charge of training of Modern Joseon Army(Byeolgi) from 1880s, which caused Im-o mutiny(1882).

Some politicians were targeted or killed by, well, cool guerilla guys like main characters for collaborating with foreign powers, and few of them actually survived long enough to get their dirty reward from Empire of Japan after the fall of Joseon.

It seems there were some politicians who actually believed that policy would save Joseon though. There are some pro-Japanese revolutionaries attempted coup but failed, fled the country(Gapsin Coup, 1884) to initiate fast modernization and stuff. They were selfless patriots, even both Japanese and Korean people admire them until now, but they actually believed Japan is ally! Meh.

You can google the name of events or persons for more info.