Matthieu_s Your EA-purchased pack should be usable with your Steam install. Normally you'd open the App, click Sims 4 > Manage > Manage add-ons, and download the pack from there, and the content would install into the Steam version's program files. However, if you also installed a second copy of Sims 4 through the EA App, the pack is likely there too, which is why it wouldn't load.
The fix is to uninstall the game through the App and reinstall only that pack. More details are here, in the "I have a Steam install" section of the first post:
[CURRENT ISSUE] Sims 4 DLC shows as installed, won't load in-game | EA Forums - 11831673
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