Will I be able to get the free Pokemon game if I link that same 3DS that I’ve unlinked from Club Nintendo a while back

The game should have come with a leaflet that has the code on the back. You're not completely out of luck though, because I found this in the FAQs on the Club Nintendo website:

My Game Didn't Come With a PIN Code. How Can I Register My Game?

Answer:

If the game you purchased is eligible for registration and the card with the PIN code is missing, please contact us and provide your name, username, email address, and the game title you're inquiring about.

That question is on the North American Club Nintendo. You can select 'Other' > 'UK & Ireland' for your country, but if you're using the UK Club Nintendo website you should contact support with this form.

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.