Spent years fighting kitchen lines packed solid with old grease before I figured out the trick is mostly about the right head, not raw cable size. My small hand spinner just punched a hole through the mess and the clog came back in a month. What finally worked was stepping up to a proper drum machine with 3/8 cable, since the bigger cable lets you swap heads easily and has the torque to chew through that hardened stuff instead of skidding past it.
For the head itself, go with a drop-head auger bit or a greaser-style clog cutter. The spade heads that work fine on hair soap clogs just bounce off grease. You want something that scrapes the pipe wall as it turns. Run it slow and work back and forth a few times rather than just powering to the blockage once. After the cable run, yes, flush with a hose bladder if you have decent water pressure. It washes the loose sludge off the walls that the cable leaves behind. Skip the chemical gel openers, they mostly just sit on top of grease and make a bigger mess. And tell the customer cold water and no bacon fat down the sink, or you will be back in six months.