Your policy should be enough as long as the Agent knows that you are renting. You should also make an effort to correct any known hazards on the property and then you have done what you can. You also should require that tenants carry a policy and name you as an additional insured if you are still concerned and that is normal in commercial leases but not in residential units but I have seen it.
Only If and until you and I sign an Agreement for Legal Services, I am not your attorney. These answers are provided for informational and/or novelty purposes
Your problem is (unfortunately) becoming more common. Assuming you are willing to personally guarantee the mortgage, there is no sound reason for your lender to refuse to allow you to title the property in an LLC. Having a blanket policy against allowing LLC ownership is a way of forcing you away from a usual means of holding title, just so the lender will not have to train its people on the subject of LLCs and how to lend to them.
Your instincts and the answer you already have about your umbrella liability policy are both also correct. It is likely that insurance is your best protection even if you could use an LLC to hold title.
If you decide to transfer title to an LLC after the loan closes (which many lenders actually advise their borrowers to do) you will always live with the concern that your lender will learn what you have done and "call" the loan. Nonetheless, it is probably unlikely that they will concern themselves once the loan has closed.
Have you tried searching further for a more enlightened lender? That might produce the best possible result - LLC ownership, umbrella liability insurance, and no worries about the lender once the loan is closed.
Answering your question on AVVO, does not create a lawyer-client relationship between us. Under the rules of the Supreme Court of Illinois, or the rules of other jurisdictions, this answer may be regarded as advertising. Because questions provided on AVVO simply cannot contain a complete description of all the relevant facts, information contained in this answer should not be considered as individual legal advice or legal opinion. You are urged to consult an attorney licensed to practice in your State regarding your own legal situation.
I agree with both of the prior answers, but wanted to bootstrap a bit onto their answers.
Regarding known hazards, Mr. Alexander is absolutely correct. However, to be vigilant you might want to find a handyman you trust and have him/her do some kind of an annual inspection on the property to make sure that no new hazards are presenting themselves, or so that you can immidiately address the ones that show up.
Also, as Mr Hanson says, you might wish to seek a more enlightened lender. I can tell you from past experience that there are a number of mortgage brokers and lenders who simply don't know how to deal with the LLC issue in rental properties. However, there are also those who do.