Since you have your movable.sed, you can open your SD card backup with ninfs. (You will also need boot9.bin, but you can dump that from any 3DS or just find a copy floating around the Internet - every copy of boot9.bin is the same.)
ninfs will decrypt the files, but after decryption you will still have to unpack the decrypted files with 3ds-save-tool before you can use the save data with JKSM or Checkpoint. (While you can use ninfs to re-encrypt them for your new SD card on your new console, you will still have to use Godmode9 to fix the CMAC of the re-encrypted file, and a few games – mainly Animal Crossing and the Pokemon games – use anti-save-restore, so it is easier just to use 3ds-save-tool.)
An alternative to ninfs and 3ds-save-tool is save3ds, but it is not easy to work with as it has no GUI. (But then, neither does 3ds-save-tool.)