This of course, is not legal advice, just an example of how i've run things. I dealt with this for years, and its frustrating. The only true, legal and private option is to pursue LLC insurance and LLC financing, which, as you say, is difficult and more expensive. A cheaper and mostly valid way to get around it is to buy, finance, and insure it in your name. Then quitclaim it into an LLC. The bank could have a due-on-sale clause, but they will never notice your quitclaim. As far as insurance, I've never had a problem insuring the home in my own name, and then just leaning on my LLC's umbrella policy (which you also should have), if it comes to that.
As long as we're on the topic of LLC's, one other caveat is if you ever go to court. The judge (in Michigan anyway), will not allow you to represent your LLC, even on simple landlord-tenant issues. You need to hire an attorney, or file the evictions/summons as yourself, which kind of defeats the purpose of an LLC in many ways.
So if you want true LLC protection and privacy, you need to pay all the way, LLC financing, LLC insurance, registered agent, and attorney representation. If you want the impression of protection (which is a worthwhile option), just quitclaim the property into the LLC once everything is setup. Again, this is not legal advice, just an example of how i've run things.