OP will not able to do working holiday in Japan. Japan has working holiday system for people from 26 countries, and the U.S. is not one of 26 countries. As OP indicates his residence as the U.S. I think simplified generalization for OP is:

1. Find the skydiving operator in Japan to help OP get work visa in Japan.

2. Need to have commercial pilot license issued in Japan using OP's commercial license from FAA. This also requires help from the skydiving operator in Japan, one of many reasons is the form used to get a commercial pilot license issued in Japan using FAA license is in Japanese. Also, the form is based on Japanese aviation policy and can be different from FAA policy. Need a person knowledgeable in this process to translate your information from your FAA license, including your flight log, into the format of Japanese regulation,

I think Step 1 is more easily available if you Google, because Step 1 is not specific to commercial pilot. Step 1 applies to anybody who wants to get work visa in Japan.

Step 2 is more specific information. For non-Japanese pilots fly on Japanese airlines, (ANA, JAL, Peach etc.), the airline will help them on this process of getting a commercial pilot license in Japan and ANA, JAL, etc. are experienced in that process. There may be website in Japanese who explain this process, but not sure how much information about this you will able to get on English website.