Well it doesn't look that tough. That rectangular scab patch is UGLY. Got to go. From the large photo of the damaged area I would cut out the piece along the grain and cut out the entire piece in the routed channel. This would hid the lines better.
My guess is this is a lacquer finish with hide laminations.
Steps:
1. Get some DNA and check to see if it dissolves the finish. If it does, you have shellac.
2. No luck on the shellac, put some lacquer thinner on it and see if the finish dissolves.
3. Clean the surface of finish where you will do the repair.
4. If its hide glue, take a clothes iron, moist cloth and heat the repair section and soften the glue.
5. cut out the bad section and begin the repair.
Joe this is a pretty big job on the skills list if you want it to look good. If you haven't done this before, you might need some "adult supervision" on the repair project.
good luck
dan