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The EXP DLC wasn't that useful by the time it released. You can add every unit to the production list way before you beat every stage. Except for a couple added by beating the game and finishing all the quests.
And, of course, the trophy is inaccurate - you only have to add every unit to the profile, you don't need to add everything to the production list. That means you only need to see every unit, not collect them. But I learned that by collecting everything.