This is a great question and so I want to attempt an answer (which will probably be something less than satisfying).
I think that the positive project of accumulating power and wealth pervades the entirety of capitalist society. The Ruling Class is the most successful in realizing this project. As the Ruling Class (the conquerors so to speak), their positive project is to more specifically secure their power. The first step in this project is to ensure that power and wealth accumulation is a defining and pervasive aim, that the appearance of class mobility isn't shattered, that the working and middling classes can always be gazing upward: idolizing the ruling class (or the class above them), lusting after their achievements, swooning over their successes, and happy to advance towards their position in society. The second step is to ensure that if possible at all, the accumulation of power and wealth is limited - that those who do climb up are those who the ruling class wants to climb up. The project of accumulating wealth and power is the string that ties the members of every class together: for the worker to play the lottery, to become a fool for the entertainment of others, to increase their productive capacity, to learn to manage the labor of others, etc. - this underlying project is present. It is needed as the most common of all projects so the ruled can respect those that have accumulated wealth and power, to hate them for their success, to fear their appearance of superiority, to love their etiquette and tastes, to vie for their cultural, to hope for their luxury, their travel, their stories-to-tell: ultimately... to accept power and wealth accumulation as the measuring-stick of ones existence and always attempt that impossible project of reaching such a position by "half-measures".
So to me, there is this game (this trap): and then there is struggle against it. Whether that struggle takes the form of breaking the rules set by the Ruling Class, destroying their property, organizing massive resistance, challenging the appearances (the illusions), etc. - that project isn't a class struggle because it is the struggle of classes with opposing interests: it is class struggle because those least successful in this game, with the least stake in the system, most attacked and exploited and abused by it - they are (we are) able to take a different view more easily. The view I mean is this one: that one has more of an interest in destroying this system than in reforming it, trying to become successful with-in its logic, or even simply accepting the project of capitalism as something inevitably permanent. In this sense, "class power" is an assertion that the distribution of wealth and power doesn't need to follow the logic of capital: that rejecting capitalism and its rules is the surest subversion of the Ruling Class's capacity to rule. Class power as those of a class refusing the logic of capitalism and fighting against it... not as empowerment of the working class as a whole, as a class, because of their class. In other words, "class power" as a statement to encourage those of the exploited class to fight against these systems... to use what power they can develop towards that end. Power to (self-determination) ...not power over (wealth and coercive force).