He doesn't seem to have all the memories of all of those lives. He seems to have memories related to battles. If Mat had all of the knowlege of every guy that is in his head then I would agree with you. But to me it seems that he doesn't.

Being able to speak a language does not equate to being able to read or write in that language. I have met many illiterate English speakers, and I have met many more that speak very well, read fine, and write very poorly.

Besides, Mat already seemed to have some ability in the old tongue before he got everybody's battle memories. I think it is very probable that Mat would spell poorly in the common tongue.