I have a Japanese 3DS and here's what I've done:
1. Install
LumaLocaleSwitcher
2. Download DLC or Update in *.cia format from the same region of your game if possible. If not, download a DLC or Update from a different region, if that's all you can find.
3. Install the DLC or Update using FBI.
4. Now, as you have a Japanese console and are running out of region games, to make it work properly, you need to change the region and language of the DLC or Update from "system default" using LumaLocaleSwitcher to the following configurations, depending on your case:
4.1. Game and DLC or Update are from the same region: you need to setup the region of the game and the region of the DLC or Update to match the original region of the *.cia files you have installed, in your case EUR.
4.2. Game and DLC or Update are from different regions: I'm not sure what I've done at the time, but just try changing the regions according to the region of the *.cia files and see if it works. Otherwise, try to match both with the game's region. One of these options will work.
Alternatively, you can use
LumaLocaleRegionAutoSetter. Read the instructions for more detailed information.