Yikes. I'm so sorry.
I'm sure it's not your first choice, but there are composting toilets that are legal in most locales.
Not to pry, but have you been putting RID or the microbes of some kind down there? Does your family use a lot of bleach?
Because enough bleach, even in suspension, makes for an unhappy tank, which means stuff doesn't break down....
Can you throw a couple of boxes in, wait an hour or so and then put a LOT of hot water through to see if you can melt and/or have the microbes eat the blockage?
Good luck. Not in Appalachia but live in a similar house so will probably have this problem sooner rather than later.