Why black dwarfes make sense in fantasy setting

Let's start with human evolution
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As some commenters pointed out, the evolution of light skin in humans had as much to do with vitamin D as it had with sexual selection. Extremely white skin seems to be seen as beautiful in at least three pre-colonial cultures: Europe, Japan and India (before Western beauty standards were exported to the rest of the world). Lower UV radiation in higher latitudes made fair skin less of a disadvantage and therefore enabled this trait to exist. Also, it is rather universal across Eurasian cultures to associate lighter skin with nobility (those who spend time indoors and don't work) and darker skin with poverty (those who work in the fields).

Evolving the dwarves with sexual selection
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Knowing that gives us some ideas to work from. For example, since dwarves are very different from humans, perhaps they have a preference for darker skin despite not having an evolutionary reason for liking it. Actually, if dwarves rarely leave their caves and mines but still require at least some vitamin D from the sun, darker skin would be a disadvantage and sexual selection paradoxically selects for less-adjusted individuals, because survival of their genes proves their resilience and child-rearing possibilities. Of course, this is not something your dwarfs would think consciously or even subconsciously. It's an evolutionary explanation why the "gene for liking darker skin" could persist in the dwarven population. Honestly, I wouldn't even mention this detail in the media itself, maybe in an interview or a blog post if the topic of black dwarves not being realistic is brought up.

Mixing in the aspects of the dwarven culture
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Going further with our parallel with humans, I think it's hard to justify skin not getting darker when a dwarf spends more time outside. If you want to put cultural significance on the tone of the skin, you need to use some imagination. Cultures are not very logical, so you can go wild with it. Maybe noble dwarves spend most of their time at leisure on the surface or are forced to stay outside as they perform their diplomatic duties with other races. Maybe trade-dwarves are incredibly rich and richness became associated with having a good tan. Maybe it's not about being rich but dangers and inconveniences of leaving the safety of the caves, so those who often venture outside out of necessity or adventurousness are seen as brave heroes. Maybe alcohol or a certain type of popular dwarven drug causes their skin to darken and the dwarf with the strongest head is admired. I'd like to point out, that you cannot go on a raid without going outside and if your dwarves are proud warriors, they certainly love raids. The possibilities are limited only by your imagination here.

So how are the fair-skinned dwarves still the majority?
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In your question, you want most of the dwarves in your world to be white and only a minority to be black. This is easy enough. You mentioned yourself that it doesn't make sense biologically for dwarves to have a darker tone of skin, and that's true. The thing about sexual selection is that it forces evolution in a direction that doesn't make sense and is often quite rapid compared to other evolutionary pressures. It biologically doesn't  make sense for humans to have white skin (not olive, but completely white skin like in Scandinavia). Our love for white skin is the reason why white people get sunburns and skin cancer even in the area where white skin originally evolved.

You need to spread the dwarves geographically as far away as possible, depending on the interconnectedness of your world. If the technological level of your world allows for the presence in your setting of dwarves from a distant continent separated by an ocean or untraversable terrain (like Sahara + Kongo in our world), this would be the best. Then a preference for black skin can evolve in one group and not the other. You can also do to your dwarves what likely happened to humans, but in reverse - first species of dwarves evolve far north/south with fair skin, but with a preference for darker skin, which doesn't evolve, because dark-skinned dwarves suffer from vitamin D deficiency at that altitude and have lower rates of survival. Then they migrate towards the equator and suddenly dark skin becomes less of a disadvantage, so sexual preference takes its course and, over the course of 2000 generations or so, drastically changes the skin of the population in the hotter climates. This gives us a nice opportunity for world-building interactions, where light-skinned dwarves are shown being enamoured with their darker-skinned counterparts from distant lands. Compare this with human fair-skinned maidens from our medieval ballads.

For the sake of realism, it would be best if there were distant and preferably vast lands, where dark-skinned dwarves are the majority. If you really have an important reason for making dark-skinned dwarves a significant minority in the entire world, it's better to limit their population density and keep the lands they inhabit at least comparable to the rest of the world. I wouldn't go lower than 20% of the world being the land of black dwarves, and if we assume there is a good reason for dark-skinned dwarves to exist in the first place, that's already a very, very low figure.