Operating as an LLC is much safer than as a sole proprietor because your personal finances and business finances are separated legally. You can then operate with business insurance, and personal trainer insurance, but you still need to have waivers drafted by legal experts. Then you MAY be able to find one agency willing to try to give you home insurance with an in-home business.
Really, no one is willing to do it because it’s simply too risky. Think of home many things can happen to a home and how many ways it can come about. Then think of how many things can go wrong with a business. So to try to insure all of those things with essentially one policy, it would be the most expensive policy you’ve ever seen, and probably the most extensive set of documents you’ll ever deal with.
Overall, you’re better off training at other locations (such as public parks) and designating part of your home as an office area. It will allow for all the tax benefits without nearly as much liability, but PT insurance is still necessary.