I've had the exact same Mii for years. I want to keep using it. I completed the Wii -> 3DS Mii transfer steps, but I want this transferred Mii to be the Personal (and only) Mii on my 3DS. I don't want to factory reset. Homebrew can help.

Aside:
I suggest you set up ftpd for transferring files between your computer and 3DS. There will be a lot of small steps going back and forth. (Just scan the QR code on that page from FBI.) Everything below assumes you're using ftpd, but you could pop the SD card into your computer and back instead.

This old post is all I could find on the topic. It basically says you can't do this, just factory reset. Instead I took some time to figure out that it is possible. Here's how.

Extract the Mii Data

Get the 3ds extdata dump and restore tool (download mirror). Extract it. You should have a folder named "extdata_dump" with "extdata_dump.3dsx" and "config.txt" (and a few other files) in it. Edit "config.txt", so that it contains:

; Dump the (Mii) CFL_DB.dat file.
DUMP "f000000b:/CFL_DB.dat" "CFL_DB.dat"
; Restore the (Mii) CFL_DB.dat file.
RESTORE "CFL_DB.dat" "f000000b:/CFL_DB.dat"

Connect to the 3DS FTP. Transfer your "extdata_dump" folder into the "3ds" folder.

On the 3DS, close ftpd and open the homebrew launcher. In homebrew launcher, find and open "extdata_dump". Pick "Dump extdata specified in config". After it says success, and switches back to the menu, quit and reopen ftpd.

In the "extdata_dump" folder (in the "3ds" folder) you'll now find a "CFL_DB.dat" file. Download it to your PC.

Edit the Mii Data

Get the 3DS Mii EDIT Tool (download mirror). Run it, open the downloaded "CFL_DB.dat" file.

Switch to the "All Miis" tab, pick the transferred Mii. Click the "Set as personal Mii" button. Click the "Save and overwrite..." button, then quit.

Replace the Mii Data

Switch back (on the computer) to your FTP tool. Upload the now edited "CFL_DB.dat" file to the 3DS, same place you got it. Quit ftpd, go back to extdata_dump via the homebrew launcher.

Pick the "Restore extdata specified in config" selection. Wait a moment, and you're done!

Launch Mii MAKER to see your new personal Mii, and now safely delete any other Miis you don't want.