As the question reads, I'm trying to find sources for the Battle of Badr outside of the Qur'an from the same time period.
It depends what you are prepared to accept as "sources".
If you are looking for written sources, then you are likely to be disappointed. The Wikipedia article on the Battle of Badr lists the extant primary sources.
Although the Battle of Badr was the first relatively large-scale confrontation between the followers of Muhammad and those of his opponents in Mecca, it was still not what many people would consider to be a large battle. There were only about 1500 combatants, and less than 100 casualties. It was fought in a part of the Arabian Peninsula that was relatively isolated from much of the rest of the world.
Its significance was that the smaller Muslim army had defeated a much larger force. The Muslims could argue that this showed that God was on their side (in much the same way as the English did after the Battle of Agincourt), but it is hardly a surprise that it wasn't widely known about outside the region. Few people had any reason to write about it, and, as far as we know, only Muhammad did.
Of course, in addition to the record of the battle preserved in the Quran, the battle would have been preserved in oral traditions. The fact that the battle was explicitly mentioned in the Quran would have made those traditions more important. The hadiths are later collections of these oral traditions and so are also sources for the battle, with all the usual caveats about oral tradition.