As far as I'm aware, you only need two things to succeed here:
A) Retrieve the quest from the Fighters Guild. (Even if it shows up in your quest log, go back and ask about the quest again anyways. The quest book is very fickle with tracking quests correctly.)
B) Have *only* Random and Wendala in the active party. (Louis, or any other permanent stragglers will throw this off, as you simply cannot remove them until they leave on their own.)
Everything else that has been said beyond these two points are just untrue. You don't need any special items or equipment, just the quest *properly* initiated and the specific party of two. If that doesn't work for you, then I can only assume you ran into a bug, or maybe Indy has changed the quest in the most recent version of LF1?
You might just try holding off till LF2. You can still initiate this quest somewhere in chapters 6 or 7 of LF2, and it should have no issues at that point.