The Showdown textures go in a folder named Textures. That folder should appear right after you click on pj64 2.0.
I had a problem with directories myself when I tried to set it up. This is what mine look like. Make sure everything is the same as mine other than the drive which is "E" for me but may be different for you. Other than that the Project64 2.1\Plugin\ ... \Save\ ... \Save\ ... \Screenshots\ ... \textures\ should look the same. I have 2.1 if you have 2.0 you probably need to change the 2.1 in my screenshot to 2.0.
Keson in his Showdown thread has a link that shows his emulator settings in his first post. Copy those.
WAIT A SEC. I'm confused by your 1st post. Showdown 64 only works in pj64 2.0, 2.1 maybe 1.7. IT DOESN't work in 1.6
Invasion ONLY WORKS in 1.6. The textures file names are different and not compatible. If you re-dumped and re-named all the textures you could play it.