China was not a unified country at the time. Although you're right in saying they suffered heavy losses, China did not so much win the war as in they didn't lose it. What I mean by this is that once Japan surrendered, there was a lot of previously occupied space that was suddenly vacated instead of won through war (thus giving one party an advantage over another). The alliance between the Nationalist Guomindang Government and the Communist Party no longer had a unifying enemy to keep them together and they resumed the hostilities they had prior to war. Also the Soviets entry into Manchuria opened the door for the Communist to have a new powerful resource base from which to operate against the Nationalist who had suffered the brunt of Japanese aggression. Even though China was given a seat in the Security Council, this was more symbolic than anything because the Nationalist (to whom that seat had been given to) first had to defeat or bring the Communist back into the fold before they could do anything else. So long story short, China had a lot on it's plate at home that prevented it from acting in international stage.

Although I suspect the United States role in the War Crimes Trials having a lot more to do with them getting to Japan first, I'll leave that question to someone more knowledgeable on that topic.

The trials were held under the control of the occupying authority, that being MacArthur as Supreme Commander for the Allied Powers. Even if the US had agreed, China was in no condition to be the occupying power. They were in really bad shape, and couldn't have supported a occupying army in Japan if they wanted, let alone feed the Japanese. Rebuilding Japan would have been totally out of the question. And Chinese occupation would have likely gone very, very badly for the Japanese.

The Chinese were also fighting a civil war between Chiang Kai-Shek's Kuomintang and the Chinese Communists, although it had been diminished during the Japanese occupation, it heated up again in 1946. So they would have never agreed on who was going to occupy Japan and with who's army. Note that Japan did not sign a peace treaty with Communist China until 1978.