Would a "basic metabolic panel" show liver problems, in non-alcoholic?

No. It may show abnormal sodium levels but that is very non-specific. You need the liver tests which is in a comprehensive metabolic profile.
Yes, it could. A basic metabolic panel includes glucose. If it is elevated, it could indicate diabetes, which is one of the most common metabolic abnormalities. It could also point to a kidney abnormality as electrolytes are part of the profile. It won't show liver abnormalities however. That would require a hepatic panel or a comprehensive metabolic panel.
CMP PANEL WILL SHOW. Cmprehensive metabolic panel will include liver enzymes, bilirubin and alkaline phosphatase and is the test for screening liver problem or just hepatic function profile if you want to narrow it down.
Not included. The limited number of tests included in a basic metabolic panel doesnt allow for liver enzyme measurements. Your doctor could order those tests individually or get a few of them automatically with a comprehensive metabolic panel.