I wouldn't try it on a piano you value. If you could get something to practice on, do that first.
I have not used this kit. The poly that I use is made by Konig and comes from Allied Supply in Pennsylvania...Ruth and Webb Phillips. I think that this is a different "Allied". Be aware that there are different types of polyester. The type they spray on a piano uses a liquid hardener and takes longer to cure. That's what is offered in this ad. The type we usually use for repairs has a paste hardener, and cures quickly. Either will work, but they are different to work with.
There is alot of potential to make things worse when it comes to polyester. For one thing, even with a small repair, you end up sanding and polishing out a fairly large area. You have to get the surface you are working on completely flat and smooth. Any scratches you put in with one grit of sandpaper will need to be polished out with a lighter grit. While it isn't impossible to do, it is tricky enough that I would be cautious. In general, a clear finish is harder to repair than an ebony finish. You have to get the wood color to match. I haven't seen the damage, so I don't know how bad it is. If it is down to the wood, and you have to stain, then it gets tricky.
If it is just a crack, you might be able to fill it in with CA glue. CA glue will bond to polyester. For very small repairs you can build up CA, and polish it out. Good Luck.