1300 is probably too late to solve Japan's internal issues and lead it on a path toward outside expansion (first into Ainu territory), but if somehow you had a successful Kenmu Restoration, maybe Japan might expand to Hokkaido (although more likely the empowered imperial court would spend their time smacking down domestic enemies). From Hokkaido, Karafuto, and Chishima, there might then be expansion into Kamchatka, then Alaska, then down the West Coast of North America, and maybe by the end of the 16th century plenty of settlements in the area. From there you need this nation to follow the path of the United States, but somehow even more lucky in the early days, and eventually expand to cross the Rocky Mountains (and defeat French/whoever fur traders and their Indian allies) to conquer the entire Mississippi basin (limiting European settlements to the East Coast and Caribbean). And like the United States, somehow remain one nation in the process. Maybe Japan colonises Australia while they're at it. Meanwhile, the Japanese have expanded deep into Siberia, meeting the Russians somewhere around Lake Baikal and the Yenisei basin. They've also colonised Taiwan, the Pacific nations, and the Philippines, and have dominance over much of Southeast Asia and a China divided between several competing dynasties (including a Manchu dynasty which has ceded several areas along the rocky coast of Outer Manchuria for Japanese settlement in exchange for economic/military assistance). Later, Japanese explorers and merchants play a major role in colonising Africa. Much of Mesoamerica is made up of Japanese client states, as is the Andes.
So the end result is that the economic power of these Japonic nations leads to Japanese becoming the world's lingua franca. Not particularly plausible (although maybe bits and pieces might work, and a Heian period or earlier POD would be much more plausible), but not entirely ASB either.
Other than Japanese, maybe have some South Indian state follow in the footsteps of the Chola, but wank them like with that medieval Japan wank above. Then you might get the Tamil language (or potentially another Dravidian language, but Tamil is most likely) as a lingua franca. Maybe have some sort of Hindu-derived religion emerge in this empire which encourages foreigners to convert (although it would probably still use Sanskrit as the liturgical language).
Or have a Finn-wank. Somehow have the Finnish language in medieval Finland exert itself more, and also have a Kalmar wank/united Scandinavia which colonises the Americas before anyone else. The colonisation is dominated by Finns, and eventually the New World (Vinland/Viinimaa) is recognised as a "New Finland", with European administrators expected to be fluent in Finnish. Eventually they revolt for their independence, eventually producing a state consisting of the territory of the modern US, Canada, and northern Mexico, plus some Carribean islands. Aggressively expansionistic, they reduce the rest of the New World states to client states, and have colonised large swathes of Africa and all of Australia. Their influence stretches globally, and the huge military-industrial complex of the Viinimaan tasavalta is the main force protecting Europe from the communist Soviet hordes (or perhaps the other way around?), after Vinlandic intervention in both world wars. Given the Vinlandic lead in the tech sector (including the Internet) as well as their nation's dominance in trade, industry, and global influence, it is highly recommended to learn Finnish in this world.
Overall, the prospects are pretty dim for a non-Indo-European language to be the lingua franca post 1300 AD. Nothing but an utter wank of some culture can possibly work.