As far as weaknesses go, Tony's Quarry is on the mild side. 4 fight would be an issue for many investigators, but Tony fights at 5 right out of the gate and often goes way beyond that. 3 health is pretty average and means that most set-up investigators can take care of this monster in 2 or even 1 action(s). An evade of 1 is laughable and 1 damage is standard. 2 horror is a bit scary since Tony has low sanity but as he can reasonably kill this in 1 turn, that often won't be an issue.
The biggest problem with this creature is that it spawns with a doom on it. That, admittedly, is pretty bad. Worst case, you draw your weakness with not enough time to go hunt it, and it costs you AND your fellow investigators an entire turn by prematurely advancing the act. Also, depending on the scenario, "furthest away from you" could mean in a place you can't reasonably get to.
However, since TQ gets a (free) bounty on it, you will have a bit more leeway in going after it, since you can engage or fight it for free. This also means that killing it gives you a free resource for your trouble. And if the act was about to advance anyway, this is essentially a monster that doesn't do anything beyond camp a location.
All these things considered, I think this signature weakness is not one of the weakest in the game, but it isn't far off. I would be more concerned with Tony's sanity and low willpower if you were to play him!