From today’s 5.4.8 patch notes:
Going to Need a Bigger Bag is now an account-wide achievement and items obtained by all characters on the account now contribute credit towards completion. This means the achievement can be completed by obtaining all the required items across multiple characters instead of having a single character obtain all the items needed.
Now I can work on this on whichever alt I bring to the Timeless Isle. I don’t suppose it gives you credit for what you’ve already picked up previously though. But still … yay!