There are ways to fight Dementors other than a Patronus.
When Harry has to write an essay for Snape, he’s fairly sure he’ll get a bad grade since he doesn’t agree with him on how best to fight Dementors - proving that there have to be other ways. If the Patronus Charm was the only way to fight Dementors, there’d be no possibility of disagreement.
“Harry fully expected to receive low marks on his, because he had disagreed with Snape on the best way to tackle Dementors, but he did not care: Slughorn’s memory was the most important thing to him now.”
- Harry Potter and the Half-Blood Prince, Chapter 21 (The Unknowable Room)
So, there are other ways to fight Dementors than the Patronus Charm, so Death Eaters could use another method. It’s simply not true that a Death Eater would never need to fight off Dementors. They’re only the Death Eaters’ “natural allies” because the Death Eaters provide them with more victims when they’re in power. However, Death Eaters are not immune to Dementors, and Dementors don’t care much about the moral alignment of their dinner.