I expect this to be normal. I assume that, in the normal usage without Pebble, the music would also play as long until it finishes or you open the notification and read the SMS, right?

Some details about the background:
SkippingStones just forwards the notification.
It does not mark the notification as "read" or "accepted".
Hence, the phone doesn't know anything about what SkippingStones did.
So, it just performs its normal task for when an SMS was received.

One option to change that would be to cause SkippingStones to mark/clear notifications once it receives them.
Another, more complicated one, could be to wait until the SMS view was closed on the Pebble and then silence the notification.
However, this requires that the watch sends a notification about this and I don't even know if it does so.
So, this is just a very wild guess.

But well, without wanting to sound repetitive, I will likely not have the time to look into this.

I am also voting for merging the threads. I explained my reasoning in the other thread.