New 3DS microSD Management – connecting from Linux

The answer is you can change SD cards as long as you are willing to move the data to your new card, and you will not lose any data.

Here is a link to the official nintendo guide to migrating SD cards.

You just have to copy stuff over. Also make sure your new card is formatted as FAT32, or you could just use your 3DS to format it.

Unfortunately, as you have noted, without the original system to transfer to, you are likely without hope. 3DS titles are tied to console, not to any ID (Nintendo Network ID or otherwise), so without the console, you can't do a whole bunch. If you had the console, you'd be able to do things differently, but as it stands, there isn't much you can do.

There are stories online where people say you can contact Nintendo to get your stuff back, but since Nintendo themselves seem pretty hush on the matter (their website states that you can only do things through transfer), I don't know how lucky you will be. Club Nintendo keeps a sort of record, in a way, so they might be able to help that way.

I would also keep in mind that stores usually wipe any used consoles they receive, so even if you can get to the store, it is not necessarily true that you can even get your stuff back, even if they have your particular console still, as doing a format wipe of the SD card and the system means that your information will also be gone. This prevents the next owner from being able to essentially get free games by downloading what you had already purchased.