We are on a septic. After a large rain - 3 inches or more in a 8 hour period - the tub backs up with water (dirty) and clear water flows out from underneath the toilet. Is there anything we can do to resolve this problem. The downstairs bathroom is below ground. Thanks.

Sounds like you need a sump pump

I am going to pass on the sump pump for now because your description has the water inside your septic system and not from the foundation. Your septic tank is becoming water logged and flooded. Is there any way for you to alter the drainage of the yard where the field tiles are located?