004:
Yeah, it is from the Pokémon Patchin (ポケモンバッチン) product line. 
Let’s start with the official website accessible through the internet archive.
https://web.archive.org/web/20030320161156/http://www.patchin.jp/
If we navigate to the products page, we can see the first products listed back in 2003, and the Media Factory Customer Support Center information below it.
We want to know which release it is from. Thankfully, there is a fan made website documenting them (incomplete).
Using the search term スターミー
You can reach the following results:
Note the number on the result title, matches the product identification number on the ‘snap’ (name for them).
Let’s click on the result.
We can see the recorded release information, being ‘snap’ 54 out of 161, coming from the 3rd expansion pack.
Now the product ID makes sense:
054(snap number) - 3(expansion pack).
Let’s find a release date.
Navigating to the archived version of the Media Factory website, we can find exactly that.
8th, October, 2004