With the Civil War synopsis emphasizing the global aspect of the SHRA, T'Challa could help illustrate the perspective of the international community. His responsibility, first and foremost, would be Wakanda's security. So, I can easily see him initially aligning with Stark in the pro-Regulation camp.
Here's how I see it...
So, Spidey aligns himself with Cap and is basically playing vigilante as is his usual MO. BP goes after Spidey (who Stark deems is flagrantly, publicly continuing to break the law) and wounds him quite seriously. BP learns Spidey's identity and realizes he's just a kid. He's a little overcome with guilt and tends to his wounds. Let's say some secret Wakandan medicinal herbs are involved.
BP's still undecided on whether to serve up Parker to Stark. They have a philosophical debate, and BP sees that things on the ground aren't as black/white as he previously thought and decides to join up with Cap's crew.
I think this would emphasize Spidey's inspirational quality while simultaneously giving BP his due by making him a prime mover that might tip the balance.
Also, this has the added advantage that T'Challa's acting alone, against the wishes of T'Chaka, who's currently still very much alive and on the throne. So, it's just T'Challa who's on Cap's side, and he doesn't necessarily have the backing of Wakandan's resources.