Are there any examples of philosophers who did (or do) neither analytic nor continental philosophy, but rather practiced (or practice) a harmonious mix between the two?
Frankly, the terms analytic and continental are not especially meaningful when applied to contemporary philosophers. Let me explain what I mean by that.
Sure, there have been and still are clearly analytic and clearly continental philosophers. No one would confuse Carnap or Quine with continental philosophy. And no one is calling Foucault, Heidegger, Derrida, or Lyotard an analytic. But part of what made things clear was that the original group of analytic philosophers specifically believed metaphysics was bunk.
In contemporary philosophy, "analytic" refers to a method built around argued claims -- with claim being meant in the technical sense as a falsifiable claim. With such a definition in hand, I would argue that analytic philosophy becomes largely synonymous with good philosophy. You can do go analytic philosophy about continental figures.
Conversely, "continental" in contemporary philosophical discourse refers either to philosophy about figures from the continent (including stretching back to say Hegel or so) or a particularly bombastic politicized method of writing philosophy. The latter is generally looked down up by analytic philosophers; the former is viewed a bit askance if only because its not in conversation.
Moving past the difficulties with the terms, there are several groups that are neither continental nor analytic due to historical groupings:
(1) The American pragmatists (C.S. Peirce, William James, etc) (2) The [catholic] personalists [though they might be continental on most definitions] (3) Contemporary Chinese philosophers (Tu Weiming, Roger Ames, Henry Rosemont, etc)
For some reason, I also tend to imagine Karl Popper as not fitting in well with either grouping -- but I have to admit that I haven't read him so I can't give a definitive answer.
So then I would describe McDowell as an analytic who writes, in part, about Hegel and phenomenology. I gather Dan Zahavi does something like that too .Similarly, I would describe Dreyfus as an analytic who writes about Heidegger. But returning to my earlier claim, the problem with trying to split things and find someone not in either is that the words "analytic philosophy" have become partially synonymous with "good philosophy" in this era. And I, as someone who wrote an ethics dissertation used people from both sides of the split, but I would definitely say that following an analytic method is better in one's own writing.