The most information we can get from the book can be found in the General Features section of the hideout description, in the box titled "What the Redbrands know" (page 20):
The Redbrands have a handful of captives in a holding area "near the old crypts," which are guarded by skeletons (see areas 4 and 5).
This only tells us that the Redbrands are aware of the skeletons (it's their trap after all). However, it is likely that at least some of the Redbrands would know the command word as well, as the trap is positioned right in front of their prison, and the skeletons might otherwise interfere with prisoner transfer.
If this doesn't seem likely to you, there is at least one person in the hideout that is expected to know such things: Glasstaff himself. Though it is not strictly specified in the book, it is very likely that he's the one who set up the whole trap, and thus would know the codeword.
As for how could your PC-s find this out? Interrogation, persuasion, magical effects, and others. While getting information that you don't ask for in an interrogation seems unlikely, a particularly successful one could yield it. For instance, if your PC rolled an Intimidation check while interrogating and got a very high score, an NPC might be so scared that he starts spewing out everything he knows (possibly pissing his pants at the same time). Or a well worded Suggestion ("Lead us safely through the hideout") spell might prompt an NPC to divulge such secrets. It's even possible to make allies out of enemies, and an ally would be incentivized to keep his new friends safe from skeletons.
Sticking to the book is handy for new DM's that are still struggling with all their other responsibilities and don't want to add adaptive storytelling to the list. But the book was not meant to be a cook book, so there are naturally some holes in the text that the DM is required to fill in. This is also good practice, as modifying the initial script in reaction to PC's actions usually leads to a much more enjoyable campaign than just going by the book.
The PC's might not enter the skeleton room at all, at which point the keyword is irrelevant. It is also irrelevant in case the players stumble upon the room before gathering any information. But it's good for a DM to know it exists so he can throw it in at any point in the game where it would make the game more enjoyable (as a reward for a high roll, or even as a new plot point that enriches the original adventure).
Princes of the Apocalypse looks like your best option. It is set close to Phandelver, has the same look and feel (same cartographer) and is for characters level 1-15. It really starts at level 3 though.
Your other options are Hoard of the Dragon Queen (1-7) and Rise of Tiamat (7-15) A lot of work required by the DM, but both nice adventures.
You could also try the Wizard's Amulet by Kobold games for a free 5e adventure https://froggodgames.com/sites/default/files/promos/The%20Wizards%20Amulet%20(5E).pdf