Just wondering about options because I have a '07 Galaxy Grey also, but in hatchback.
Maybe you could do a video of more than the 3-4 like a 1-2-3-4 to compare the sounds of the other shifts to see if it's just the video, but it sounds like a synchronizer thing to me. From my limited knowledge, the synchro makes the gears in the transmission spin the same speed as the output shaft so they slide together smoothly and the transmission doesn't grind.
Try this: Get up to 3500 in 3rd, clutch, put the stick in neutral, let clutch out, clutch back in, then to 4th, and clutch out, performing the actions as quickly as possible to not lose speed. Someone correct me if I'm wrong, but this is called "double clutching" and it performs the same function that a synchro does. (Big rigs typically have to do this, I am told by my driver friend.) If it doesn't make the sound, I'd say synchro.