On anything you lace, it should be tamped. It sets the lace in place. Notice that when you lace something up, the weaving of the lace in, out, thru, over, etc, it creates hills and valleys. Tamping smooths and levels it out as well as setting the lace.
Try it on a piece of scrap. Punch a dozen holes in a couple pieces of leather and lace them together using the lace pattern you are going to on the seat. Then tamp it down and note the difference. If your plan is to use a simple "X" style stitch, it won't make much difference but anything else will show.
Good luck. The seat is coming along nicely. I'd be a bit concerned about that parallel (spelling?) scar on the nose of the seat tho. Adds an interesting chacteristic as well as natural charm to the seat but be sure the leather isn't weaker there.