Actually, after reading John Man's Kublai Khan I believe the Mongol invasion would have failed all the same… if only because it was mostly made up of Koreans and Southern Chinese who had little motivation to fight and die for the Mongols and were quite inferior to the samurai anyhow…
What does qualify as 'successful' anyway ? Assuming they had ( with crack Mongol troops and Chinese early firearms ) managed to rout the Japanese army defending Hakata Bay, what then ? Submitting the rest of Japan would have been far too great a task, also at the time the Hojo shoguns' power was quite strong still; it's only in the aftermath, when they found they could not reward the samurai with lands for fighting the Mongols, that it weakened… so it's very unlikely, I think, that Japanese resistance would have collapsed.
Even assuming some daimyo [ was the term already in use ? ] DID bet on Mongol success and support them in order to be invested shogun, it's doubtful they could have been of much use considering the other demands of the already overextended Empire…