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But what about black holes that are not gobbling up anything? The only way we can measure their mass is by the gravitational effect they have on other objects. Consequently they could be part of the missing mass.

It should also be noted that some physicists believe it is possible that there is no missing dark matter, that the problem is that our model of gravity is at fault. Gravity may behave differently at the extremes misleading us into thinking there must be more mass than we can see.

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No. Black holes containing normal matter are fully accounted for. To explain this would require you to know General Relativity. If you do, then I will point you to the relevant sections of MTW to reread (because if you know GR you have read MTW).

These calculations are all based on GR. There are different models suggested, but as of right now, none are considered very serious contenders. Some do take the dark matter/dark energy problem as a sign that GR is broken.