I do not think you would win in court.
I think the owner should offer to compensate you in some way, because iit is good business, NOT because he is legally or otherwise bound to.
I think it is similar to " hey can you keep an eye on my cues? I have to use the bathroom". The guy will almost always say " yeah, sure". You come back and he is on the other side of the pool room with his back turned away from the your cues. Is he responsible if they were stolen?
To me the implied agreement is " if I actually witness someone trying to steal your cues and walk with them, between my shots during my own game, I will challenge them. If I happen to be preoccupied with something
else, I am sorry you lost your cues. I did not come to the pool hall to be a cue babysitter, I came to play pool. I would keep an eye on them, NOT my full attention. Likewise, a pool hall is in the business of renting tables, NOT cue babysitting. As has been said, if there is a lock and key situation and a safe rental is involved, it can be a different story. Even so, there is usually a disclaimer about liablility, or at least a limit to their liablity. They won't cover your $5000 cue, etc etc. You would also have to prove what cue was lost and document its value, than that value would be depreciated for its age and condition,etc.
Obviously if you leave your cue on the table directly next to mine, it is very unlikely that someone will take it without me noticing it. I would make every effort to keep anything from happening to it, but in the end it is your cue, and you had better things to do than watch over it during that time. I would feel terrible, and may offer any help I could, but would not feel obligated for full replacement value of said cue. It was too inconvenient for you to put it away, or take it with you, so you assume the risks with that decision.
I have in the past left the pool room in a hurry and left my cue fully assembled laying on the table. I was more than hlf way home when I suddenly realized this. I called the pool room in a panic and found that my cue had been put away safely behind the counter. It was my fault alone. I happen to get lucky.
I have also had a cue stolen right off a table when I went into the back room to check on some stock. It really sucks, but I had no one to blame but myself.
I have worked in many pool rooms, and have felt comfortable enough to leave my cues in pool rooms ( also ones in which I was not employed). I do not feel that the owner has a responsibility to take care of my cues or replace them if they go missing. In the event of a loss, I may ask the owner what he suggests to do about it, but would not demand it, or take action.
I think a fair solution in many cases would be for the owner to help chip in for the purchase of a new cue. For example if it was a predator, he could order you a new model of comparable value and agree to pay a portion of his wholesale cost for that cue ( with no retail markups). I think it would be a good compromise and you could get another cue for a good deal less than the full replacement costs out of pocket. Though it sounds like you are in good shape with a sponsor, so you are ahead of where a lot of players in a similar situation would be.
Just my take
JW