After you have finished the 120+40 stages in story mode, you'll have spelled between 2000 and 2500 words (depending on your skill and whether you tackled endless mode).
Instead of grinding out the last few hundred words, you can exploit a bug in the Vita version. The game enters an inconsistent state when certain words (reported: XERO, BLING, FARTS, FARTED; I couldn't get XERO to work, FARTED did it for me) are spelled, resulting in lots of words being added to your word count.
Thus:
(1) Back up your saved game to the cloud.
(2) Delete the game, reinstall, stop the download of the patch. Recover your saved game.
(3) Enter the first stage and refresh the grid until you can spell one of the words mentioned above. Due to your large health, the enemy will not kill you.
(4) When you have spelled one of the bad words, you get a message box informing you that the game state is corrupt and you should close the game. Don't do that, instead wait patiently, eventually the trophy should pop. If not, try again from step (3).
Thread about the bad words:
https://forum.psnprofiles.com/topic/39827-born-to-spell-trop...Update: According to user iXanon in the comments, this is not possible with a fully patched PS4 anymore. I did it with the Vita version.