As a former personal injury attorney, I can tell you that if your name is in any way related to the property (on the deed, rent checks etc..), you will be named as a Defendant. Plaintiffs don't have any information early on as to your business structure, so they will "shotgun" and name everyone who could potentially be involved to make sure they file against the correct entity before the statute of limitations expires.
I am not YOUR lawyer, but it seems to me your bigger problem is that your insurance policy does not have you (the property owner) as a named insured. Insurance companies margins are income (premiums and investments with premiums) - expenses (claims). If they can avoid paying a claim, they will. They have no loyalty to you, so if they can deny coverage because you weren't named in the policy, they will.
Piercing the corporate veil is a complicated topic, and this is a long reply already. The short answer is if you "disregard the corporate form" (i.e. don't treat the LLC as it's own "person") they could potentially ignore the corporate veil. One "accidental" payment probably wouldn't do it, but a bunch of them would become a problem.