It's a shame about Mongolia. I like that little country.
Just spitballing (I can resign myself to 'it's impossible', but hey, I'll try):
As an alternative, any way Mongolia can be more militant in the 20th Century? Mongolia survived not just because of Russian influence, but because it was a useful buffer state between Russia and China (even the Chinese could agree on that). Let's say WWI swings around and the Germans go all-out on Russia (France a shadow of its former self due to the Franco-Prussian War ending much differently). Russia splits into the Baltic States, Poland, Belarus, Ukraine, Finland (+ Karellia), a Hohenzollern Tsar (controlling, say, St. Petersburg and northern Russia (think, Novgorod)) under German domination, the Romanov Tsar surviving in the east as Tsar of Siberia, and then in the middle a Soviet Union made up of Central Asian steppe countries and parts of the old Russian Empire? Mongolia picks up a piece here and there and a slight population boost of Kazakhs and Turkmen escaping the *USSR (obviously butterflies change that a little), and Monarchists who see no hope in a Siberian Empire.
In this same war, Britain isn't involved, so Japan sides with Germany and together they attack China (who tries to get rid of foreign basing rights to avoid being involved, ironically). Britain later joins in on the fun (perhaps to do with the Ottomans, or threat of Hong Kong being occupied, etc.), making 3 powerful countries invading China, so then Mongolia, as with Russia, picks up what it can, using Mongolians in Inner Mongolia as supporters (there's a possibility here). Near the end of the war a radical uprising occurs in China to try and get peace after much of China (including all of the important cities, like Beijing, Shanghai and Najing) is occupied. The Kaiser and the Japanese Emperor set up a puppet Qing Empire (think Kaiserreich) while the rest falls to the Chinese Republic. The Uighurs in Sinkiang declare independence with British support and Tibet enlarges.