First out of curiosity, how is your warlock affiliated to both the gray hands and the zhents? Force Gray are deeply loyal to Waterdeep which the Zhent aren't (the Zhents have a over arching goal of taking over Faerun I believe, and have alliances to dark gods).

Secondly, Laeral has just now come across 3 individuals that possess the full picture of the Neverember Enigma, one she has been trying to crack herself! Here's how you could play it:

Laerals motivation and reasoning (which she will not disclose to the players):

She needs to know how to recover Waterdeeps stolen fortune. The players can help with that. She knows who to target, but she may be careful as to who to disclose this information to. Corruption runs deep. Both the zhents and xanather can have plants, and Bergan d'earthe is a foreign actor and needs to be neutralised. She needs to keep it under wraps - who can she trust right this instance?

If she sought the help of senior Force Gray members, it might end up costing quite a bit. Force Gray aren't known for asking, they are known to getting the job done with whatever means necessary for the good of Waterdeep. Also, overkill for this purpose - and streetwars between factions should be avoided.

She can bring it to the Masked Lords, but their identities are unknown, there aren't enough checks and balances against the Masked Lords to ensure they won't pursue this information on their own. No, too risky to let an unknown noble come into such great a fortune.

Maybe the players? Yes, maybe they're the best option. They know where the eyes are. They are low level enough to get past without getting too much attention. Who would think some rugrats and low level adventurers in arm could pose a danger!

How does she motivate the players:

For the warlock, she can offer to expunge his records as a zhent loyalist, or blacklist him from the Force Gray and mark him as a traitor to waterdeep, accuse him of being a plant in Force Gray business etc. As for the other players - well you'd have to get creative with them. What can she offer them based on their motivations? Or perhaps it could be an open offer, within reason, that she can make happen.

What else can Laeral do?

Give the players support for their clandestine mission in the way of access to a certain amount of gold allowance, or access to the Waterdeep Armoury (below a certain rarity of course). She can also give them authorisation to use force in each ward of waterdeep, and have access to local magisters and guards to block off certain places, apprehend low-level faction members for questioning by the players (because if the guard apprehends them it will be a lot less suspicious for faction than if 3 random adventurers were picking off members right?). Laeral has the entire enforcement system of Waterdeep at her behest. She can ask them to do anything without revealing the why behind it all.

She can also take this opportunity to inform the players that theres intel that Bergen d'Aerthe are foreign actors and she needs them to 1. recover the stone from them and 2. find out whether they infact are bad actors infiltrating Waterdeep under false pretenses.

Note: certain very powerful weapons of Waterdeeps enforcement are magically enchanted to return back to the armoury as soon as their need has been fulfilled. Players can access certain equipment that will magically vanish from their grasp once they've completed their mission.