I won with Takeda on Normal difficulty, it was quite hard but not overly so. Now trying with Shimazu on Hard, and I am quite willing to accept advice at this stage...
The point is, it's impossible to beat the mongols' initial 3000 troops with Shimazu's initial troops (less than 1000), you can't build more than 1 unit a season until you conquer (easy) and develop (takes time) more provinces, and the mongols move east just too fast.
I managed to take Chikugo (the river province ) on the second season and hold it. The mongols attacked once, lost a full army (16 units), then placed a holding force and went east. Many provinces are upgraded to +100% farming in this setting, which is worth over 800 koku in pillage, so they get lots of pillage even in non-castled provinces, and non of the other clans really resists them.
By 1284, 10 years after the invasion, I had Kyushu and Shikoku and was pushing into the main island through Nagato and Bizen, with training centers in Satsuma (original), Chikugo (developed) and Tosa (took from Uesugi). The best I do when attacking is 1:3 casualties, often 1:2. The mongols had some 12,000 troops, most busy finishing off Hojo, I had 3000. And then, reinforcements landed at Hizen, and there was nothing to stop them from taking Chikugo next even though I built a port there. End of story. I can't both push forward and defend the northern provinces of Kyushu.