Soft start for fridge compresser
The linked speed controller clearly states that it is for a universal motor and only a universal motor. A universal motor is a series-field type DC motor with brushes, that has been designed to run on either AC or Dc power. It is unlikely that a refrigerator will have a universal motor. If you can not see brushes and a commutator inside, or two round insulated covers for changing brushes, it is not a universal motor.

There are variable-voltage AC speed controllers that are designed to power fans and centrifugal pumps. The torque required to drive a fan or centrifugal pump increases in proportion to speed squared. They require very little torque to start and reach their maximum torque requirement at full speed. One of those might be able to to start a refrigerator, but it is rather unlikely.

Your best alternative is probably to buy an inverter that is designed to start an AC motor that has a hard-starting load. The fridge may need its own inverter. If the inverter you have can start the inverter, buy a second inverter for everything else.