What profit is liable to country's CIT where PE is triggered?
All income. The company is considered tax resident in that country because of management.
So, should US LLC pay CIT in this case despite such US LLC is treated as a transparent entity in a country where a manager taking decisions is a resident?
Not where the manager is resident but where he is taking decisions.
Look i understand what you are trying to achieve and the only way to make it work is to make the LLC tax resident in a tax free jurisdiction or a low tax jurisdiction.