I think that in city drains they just scrape it all off. But a person can fit inside a city drain, whereas even if you're very very skinny and small, you'll never get along yours
I don't think there would be much point in just moving the fat deposits along to the tank, as then that would have to be cleaned out too. If it's really that bad, and no one comes up with the perfect solution, then the drain equivalent of a chimney sweep might have some ideas, or you could dig up and replace the pipework.
Once you've done it, then never pour fat down the drain. Let it cool then scrape it into the bin, or if it's suitable, feed it to the wild birds.