Why did the Soviet Union not "grant" Inner Mongolia to Mongolia after World War Two?

Actually some of the answer is found on Wikipedia, but in the pan-Mongolism article:

In 1943, the British Foreign and Commonwealth Office predicted that the Soviet Union would promote the idea of a Greater Mongolia to detach China's Inner Mongolia and East Mongolia from Chinese influence.[46] A year later, the then-Soviet satellite Tuvan People's Republic was annexed by into the Russian SFSR. During the Soviet invasion of Manchuria in August 1945, Outer Mongolian troops occupied both Inner and Eastern Mongolia, and Japanese collaboratist leaders like De Wang were kidnapped to Outer Mongolia to be inculcated with pan-Mongolist ideals. Perceiving an imminent threat to China's territorial integrity, Chiang Kai-shek signed an agreement with the Soviets during the Mongolian occupation which gave Chinese recognition of Outer Mongolian independence. In return for the fulfillment of this longtime Soviet foreign policy goal, the agreement stated that Mongolian independence would only be effective "within [Outer Mongolia's] existing frontiers". The Outer Mongolian troops subsequently withdrew from China.[42] In 1947, Chiang renewed his claim on Outer Mongolia in response to alleged Mongolian incursions into Chinese Xinjiang during the Pei-ta-shan Incident.[39]

So there was a (little known, perhaps) deal between Chiang Kai-shek and the Soviets which exchanged Chinese (ROC) recognition of (Outer) Mongolia seemingly in return for the latter dropping any further territorial ambitions in [the rest of] China. Although Chiang Kai-shek did consider the deal void a couple of years later, there was apparently no substantial Soviet/Mongol retaliation before the PRC/Mao came to power. It's not clear just from that snippet if this lack of post-1947 Soviet/Mongol retaliation was because of time constraints or other considerations.

Frankly Wikipedia has a very confusing article on the Pei-ta-shan Incident aka the battle of Baitag Bogd; sporadic fighting seems to have lasted for a year (into 1948), so it's possible I'm missing something. But what seems rather certain is that Stalin apparently did not wish to broaden that conflict.

As an interesting tidbit, Stalin secured approval in general terms from the British and the Americans (for his plan in Mongolia) at the Yalta conference in February 1945 as maintaining the "status quo" in that region. And we have a glimpse/summary of what Stalin said to the Chinese delegation during the subsequent bilateral negotiations:

In the course of drawn-out, bitter talks in Moscow between Stalin and the head of the Chinese Executive Yuan T.V. Soong, the Soviet leader raised the prospect of Mongolian nationalism: “If this [China’s recognition] does not happen Outer Mongolia will be rallying point for all Mongolians. It’s to the detriment of China and us.” Stalin pulled out a map to show how Mongolia was strategically important to the USSR. He cited Lenin’s parting with Poland and Finland in the wake of the Bolshevik revolution as an example for China. He referred to the Yalta agreements, arguing that China, in the end, simply had no choice. With northern and northeast China poised to be overrun by Soviet troops – who were on the verge of entry into war against Japan – the Chinese knew that this was not an idle threat.

Under pressure, China’s leader Chiang Kai-shek gave in, cabling Soong to make the “greatest sacrifice” on the condition that Stalin would abandon his support for the Chinese Communist Party and the Uighur independence movement in Xinjiang. Stalin agreed. With the signing of the Sino-Soviet Treaty of Friendship and Alliance on August 14, 1945, China promised to recognize Mongolia’s independence but only after a plebiscite. Both sides realized, of course, that the referendum was just a show but Chiang wanted it for domestic political reasons, and Stalin was happy to oblige.

Consequently an official referendum of independence in (outer) Mongolia was held on October 20, 1945.

Some of the pre-Yalta steps that Stalin took are also interesting, i.e. how he pressured the allies (with respect to Mongolia) with "facts on the ground":

The Mongolian leader Choibalsan visited Moscow in January 1944 and was clearly told that the wartime cooperation with Nationalist China against the Japanese was at an end. Stalin immediately offered to fund and arm a Kazakh nationalist bandit named Osman to loosen Chinese control of the Altai region, lying between Mongolia, Kazakhstan, and the Chinese Northwest. Choibalsan would personally deliver the several hundred rifles and submachine guns with 100,000 bullets. This was also a signal for Choibalsan to begin efforts to build a Greater Mongolia at Chinese expense. By March, Chinese patrols were in retreat and the American press began to warn against border incidents endangering the joint war effort against Japan. Stalin made clear to the Chinese that the border regions with the Soviet Union would not easily return to Chinese sovereignty after the war. The American and British Allies would also have to take into consideration Stalin’s influence on China in Central Asia in balancing objectives at a time when pressure was being exerted to get the USSR into the war against Japan and to reach a settlement on Poland.

In late 1944, Stalin began to receive cables about a Muslim uprising in the Yili District of Xinjiang on the Sino-Soviet border. In February 1945 at Yalta, Franklin Roosevelt promised Stalin US support in negotiating a new treaty with China that would include the price of Soviet participation in the war against Japan, namely, joint operation of the Chinese Eastern Railway and Soviet control in the ports of Port Arthur and Dalian, together with Chinese recognition for the first time since Genghis Khan of the independence of Outer Mongolia. Roosevelt’s promise guaranteed that the Chinese would come to Moscow to negotiate, but additional pressure would be necessary to achieve best results. In the end [...] only the USSR’s entering the war could compel the Chinese to settle and sign. [...]

And Staling used similar "facts on the ground" to put more pressure the Chinese thereafter:

In March [1945], secret Soviet aid to the East Turkestan Republic in Xinjiang began. On June 2, encouraging discussions regarding future aid were held with the Republic’s leader Shakir Khodzhaev, also known as Alihan Tore, who was overseeing the drive to take Urumqi from the Guomindang forces. Stalin continued to receive regular updates until September of separatist activities that would have been deeply disconcerting for Chinese statesmen.

It's also interesting how the latter (ETR) ended up after Stalin no longer saw them as useful:

Having accomplished larger goals from Mongolia to the Pacific, Stalin’s support for the East Turkestan Republic and Osman quickly dried up. Alihan was pressured into October 1945 negotiations with the Chinese much against his will and later “evacuated” away to Tashkent in June 1946, shortly before the last meeting of the “Temporary Revolutionary Government of the East Turkestan Republic” that he had headed. The KMT would continue dismantling the ETR legacy in 1947, but by 1948 the KMT was in retreat and ETR veterans were mobilized to take power, only to be pressured into submission again, this time to a CCP delegation arriving from Moscow, again flying in Soviet planes. Many of the leaders of the ETR died in a mysterious plane crash near Irkutsk, while on their way to the proclamation of the People’s Republic of China in Beijing.

According to one account, the article/chapter of Christopher Atwood titled "The Second Partition of Mongolia" in the book Mongolia in the Twentieth Century, in Inner Mongolia itself, after the Sino-Soviet agreement:

Inner Mongolians began to agitate against their reversion back to Chinese control.

It goes on to recount the [almost totally obscure] story of Danzangombo, a former lama turned communist, who organized a small Mongol-unification militia (~100 men), under red banners, in Inner Mongolia and of a similar group the Young Men's Party. Because of his past Danzangombo was judged unfit in this role by the Soviet and Mongol authorities and they basically kidnapped him to Ulaanbaatr. However the delegates that Danzangombo had managed to assemble, proclaimed a People's Republic of South Mongolia.

The delegates clearly had Soviet and Mongol support for the formation of some for of de facto government [...] Choibalsan [the leader of (outer) Mongolia] honored the new government with a personal tour of inspection in Sonid Right on September 16-18.

Actually Wikipedia has short page on the short-lived Inner Mongolian People's Republic as it calls it, which these groups basically declared. That story ended with Chinese Communist Party representatives taking over the self-declared government of this region, after they obtained approval to do so from the Soviets in a visit of Yun Ze to (outer) Mongolia, who then went to the self-declared southern republic to implement the CCP takeover. (The Wikipedia page on this obscure Inner Mongolian republic seems entirely based on contemporary Chinese sources, so it doesn't mention these details, found in Atwood's article.)

Another amusing, perhaps, factoid from Atwood's article is that after the Mongol-Soviet relations soured because the latter didn't support Mongol reunication, Choibalsan only travelled to Moscow by train because "trains were safer". (In view of how the ETR leadership perished, that's perhaps not surprising.)