The reason that we are talking about dark matter is that it is detectable by its gravitational effects on ordinary matter. Our ways of looking at it are still incredibly blunt but they're much better than what we had 20 years ago.
We can make hypotheses about dark matter and conceive observations to test these hypotheses.
As for dismissing God, I don't see why, we just need to come up with some signature that would differ between a designed universe and an accidental one and apply for observation time on a suitable instrument.
Dark matter has a different cognitive status than God, because there is some evidence, albeit inconclusive, that suggests dark matter (from what I understand; IANAP). In contrast, there is no evidence that suggests God, so a claim that God exists would be arbitrary and is therefore rightfully dismissed.
I would say that God is exactly what you are studying when you are looking at dark matter. God is a metaphor for the unknowable ground of being, the mystery that holds us all. The wisdom traditions have been about understanding the patterns of the Universe, in all senses and to provide a mechanism and context to dissolve the idea that something exists as separate from anything else. Theists and Atheists both seem to get it wrong when they argue over the "existence" of God, as Joe Campbell said, it's about understanding how to read a metaphor and experiencing the direct internal subjective experience of reconnection.
Isn't it the other way about? The reason we are hypothesising dark matter is because we can't explain some gravitational measurements. Did someone postulate dark matter particles, then later say 'well if these exist, they must cause some strange gravitational patterns.' ?
Yes, dark matter is of course an explanation invented because of observations. But it would just be a just-so story if you couldn't probe further consequences of that explanation by making new targeted observations.
As for dismissing God, I don't see why, we just need to come up with some signature that would differ between a designed universe and an accidental one and apply for observation time on a suitable instrument.