Is the dealership at fault if it failed to properly tighten my lug nuts and my tire fell off?
d on what you have written, the dealership is likely at fault, and therefore liable for any/all repair costs: it is negligent, or unreasonably careless, to not tighten lug nuts when replacing a wheel, and when someone negligently causes or contributes to damage to your property, they become liable for the costs to repair such damage. If they will not voluntarily honor their obligations, you could sue them (e.g. in small claims court, as your own attorney or "pro se" to reduce costs) for compensation.