How to permanently remove a previous Mii from a 3DS when it is not in the Mii Plaza

Hopefully, this describes what you want to do. From Nintendo:

How to Change the Mii Character Used in StreetPass Mii Plaza

Important Note! Only a Mii character that was created by the user can be used as the display Mii.

What to do:

Select StreetPass Mii Plaza on the HOME Menu, and tap "Open."

Highlight your Mii Character and press the A Button.

Select "Mii Settings."

Select "StreetPass Setting."

Select "OK."

Choose Mii from the list of Mii Characters. Select "Yes" to confirm.

Note: This is the only Mii you can change - you can edit your personal Mii, but you cannot delete it or switch it for a different Mii you have created. (It actually warns you about this when you create them, and also in the manual.) Also, if you attempt to delete them, you get a message that says "Your Personal Mii cannot be erased." You can re-format the 3DS, however, in order to create, effectively, a blank slate for yourself, but editing the Mii is really your only option if you do not wish to take that route.

If you want to reformat the 3DS, go to System Settings -> Other Settings -> Format System Memory (it is the last option on the 4th page.) The following information will be lost after formatting the system memory:

• All data in the System Memory (including Friend List, Notifications, and step count)
• System Settings data
• Built-in software save data
• All photos in the System Memory
• All other extra data (created by some games and applications)

It will not, however, delete your e-shop account.

SpotPass is the generic name for any passive Internet feature on the 3DS, just like StreetPass is the generic name for any passive local wireless feature. SpotPass features are indicated via a blue LED and blue dot on icons, like StreetPass features are indicated via a green LED and green dot on icons.

SpotPass works via scanning for accessible wireless routers when the 3DS is closed. If it finds one, it synchronizes data for any games you have SpotPass-enabled. For some games like Kid Icarus: Uprising this is basically just an "Internet StreetPass" but other games like Samurai Warriors Chronicles have used SpotPass to distribute free DLC to early purchasers. Non-games like Nintendo Video also use SpotPass to get updates.

Mario Kart 7 does add people to your StreetPass Mii Plaza population if you've played online with them, although as you noted you can't use them in Find Mii, nor Find Mii 2 unless you pay their hiring cost. I wouldn't actually call this SpotPass, but since it's a term Nintendo made up who knows if they want to call it that or not.

Nintendo has distributed other items for StreetPass Mii Plaza over SpotPass. A new Kid Icarus: Uprising puzzle panel, and a single piece of it, was sent to everyone on February 27th. And on the one-year anniversary of the 3DS's release, Satoru Iwata (in Europe and Japan) and Reggie Fils-Amie (in North America) will visit your plaza after arriving via SpotPass. They do have puzzle pieces and can be immediately used in Find Mii, and will give you the achievement for meeting a Special Mii.