"I was talking recently with someone who's done pilates, and she said you really need to build up your core strength to do it safely."

Yes, that's kind of the problem. It's like the difficulty of looking for your glasses - you need your glasses to see well enough to find your glasses. Unfortunately, I don't know of any way to strengthen back muscles without using them. And using back muscles even a little bit more than one is used to makes those muscles hurt the next day. Anyway, that has been my experience.