The only other source that we are reasonably sure to have originated from Muhammad's life-time is the Constitution of Medina, which was effectively an agreement between Muhammad and his fledgling community at Medina that was preserved in later Muslim sources. It doesn't tell us much, apart from the fact that Muhammad included Jews in his umma/community and that he was an important leader in this community. As for why the Byzantines didn't write about him, it was because the Byzantines were engaged in a devastating war against the Persians that lasted from 602 to 628. During that period, all Byzantine territories east of Anatolia were occupied and Asia Minor itself was regularly raided, culminating in the first siege of Constantinople in 626 by the Persians' Avar allies. To put it simply, the empire was a mess and it is not a surprise that no-one paid attention to a new warlord in Arabia (who wasn't even active in the most important part of it!).

We have several Christian documents written in the 630s and 640s referencing Muhammad after his death though. The first is a Christian anti-Jewish pamphlet (supposedly) written in Byzantine North Africa in 634, which claimed that some Jews were convinced by the rise of a new unnamed prophet amongst the Saracens (Arabs) as a portent of a new Messiah. The Byzantines were of course keen to ensure that Jews do not defect in large numbers to these new invaders, hence why in this pamphlet the Jewish convert to Christianity went out of his way to explain that Muhammad was not a real prophet and that Christianity was the only correct faith. The most important extract from this document is this:

Justus answered and said, “Indeed you speak the truth, and this is the great salvation: to believe in Christ. For I confess to you, master Jacob, the complete truth. My brother Abraham wrote to me that a false prophet has appeared. Abraham writes, ‘When [Sergius] the candidatus was killed by the Saracens, I was in Caesarea, and I went by ship to Sykamina. And they were saying, “The candidatus has been killed,” and we Jews were overjoyed. And they were saying, “A prophet has appeared, coming with the Saracens and he is preaching the arrival of the anointed one who is to come, the Messiah.”

And when I arrived in Sykamina, I visited an old man who was learned in the scriptures, and I said to him, “What can you tell me about the prophet who has appeared with the Saracens?” And he said to me, groaning loudly, “He is false, for prophets do not come with a sword and a war-chariot. Truly the things set in motion today are deeds of anarchy, and I fear that somehow the first Christ that came, whom the Christians worship, was the one sent by God, and instead of him we will receive the Antichrist. Truly, Isaiah said that we Jews will have a deceived and hardened heart until the entire earth is destroyed. But go, master Abraham, and find out about this prophet who has appeared.” And when I, Abraham, investigated thoroughly, I heard from those who had met him that one will find no truth in the so-called prophet, only the shedding of human blood. In fact, he says that he has the keys of paradise, which is impossible.’ These things my brother Abraham has written from the East.”

The second mention of Muhammad is a short note written on a Syriac Gospel book around 636, presumably written by a monk noting down the momentous events he just experienced. The most important line is:

... many villages were ruined with killing by [the Arabs of] Muhammad and a great number of people were killed and captives [were taken] from Galilee as far as Beth

This note however is faded and the reading uncertain, so Robert Hoyland, a historian who basically wrote a book covering all the sources for early Islam, prefers a chronicler's account written around 640 as the 'first' mention of Muhammad, which said that:

In the year 945, indiction 7, on Friday 4 February (634) at the ninth hour, there was a battle between the Romans and the Arabs of Muhammad

These sources don't really tell us much about what Muhammad did, but he was clearly an important figure to the Arab armies invading Palestine, so he definitely was a person of political and religious significance. A bit later in the 660s we have a much more detailed account from a reasonably reliable Armenian historian now called Pseudo-Sebeos:

At that time a certain man from among those same sons of Ishmael whose name was Mahmet, a merchant, as if by God’s command appeared to them as a preacher [and] the path of truth. He taught them to recognize the God of Abraham, especially because he was learned and informed in the history of Moses. Now because the command was from on high, at a single order they all came together in unity of religion. Abandoning their vain cults, they turned to the living God who had appeared to their father Abraham. So Mahmet legislated for them: not to eat carrion, not to drink wine, not to speak falsely, and not to engage in fornication.

He said: ‘With an oath God promised this land to Abraham and his seed after him for ever. And he brought about as he promised during that time while he loved Israel. But now you are the sons of Abraham, and God is accomplishing his promise to Abraham and his seed for you. Love sincerely only the God of Abraham, and go and seize your land which God gave to your father Abraham. No one will be able to resist you in battle, because God is with you.'

There are other much later Muslim sources from the late eighth/early ninth centuries, but in broad terms they match up with the contemporary evidence - that Muhammad was a merchant, that he preached a new religious doctrine and that he led his followers on raids. The details of his life is often debated, such as whether he personally led campaigns into Palestine or if his religious movement was an ecumenical one inclusive of Christians and Jews, but I see no reason to question the general outline of the traditional biographies of Muhammad.