Can I get fat dissolving injections if I possibly have fatty liver?

You should probably figure out why your liver is not happy first.
Personally I do not recommend Kybella. It is highly unpredictable and the results can be anything from unnoticeable to permanently devastating.
The inflammatory reaction is also much worse than from liposuction.
Most people seeking this treatment have a normal fat distribution and fat is not the primary issue. Most people who get Kybella self assessed and chose Kybella and go to some med spa type clinic.
I suggest first getting a proper assessment.
Most of the problems people use Kybella for are caused by the shape and size on the mandible.
Liver disease is pretty serious so get that taken care of first. If you drink alcohol the stop that immediately. If quitting alcohol is challenging then find a treatment program.
Best,
Mats Hagström MD