Sorry for the delay, I was in town when I saw your post.To pick the cars you want to compete against, select "Favorite Opponents" and go to "Register". From there you can check what you want and then choose "OK".
Then go to "Special Settings" and turn on "Limited Opponent Cars" and press "OK". The secret menu applies to arcade mode, so if you want to race the opponents you chose, go to arcade mode and then "Single Race" and once you start the race, it should have the opponents you chose.
Normally it would, but once you turn on "Limited Opponent Cars", it's pretty much up to you what you challenge.
Edit: Interesting thing to note is if you limit the cars to ones that can't run on dirt or snow and then try to race on a dirt or snow track, you will be challenging the car your driving instead of what you chose via the secret menu. This happens even with the "Limited Opponent Cars" option turned on.