speaking 1274 invasion or 1281 invasion?
If the first one was a success (much more likely since Japan was highly unprepared, suffering from acute lack of military know-how, given that the last military operations was some 50 years back), Buddhism (specially Zen) and the code of Bushido wouldn't have the long term influence (honorable 1-1 combat doesn't fit well with continental zergrush)
Second one would probably be a bit trickier to win and keep, since by that point Japan had been in period of near-permanent alert and training their local forces, so it would probably be a hotbed of revolts, and certain parts of Bushido would probably be watered out with a more Shinobi way of life, focusing on unorthodox warfare with covert operations and assassination of enemy leaders.
Nevertheless ... a Japanese loss in either of those invasions would be a massive hit on their collective psyche, since they would be denied their belief that Japan couldn't be defeated (which in turn was a significent part of their imperialistic actions up though 1900's)