Excuse. To me it looks like a white top with purple band.
That's an odd one - pretty sure it's post-war, and likely not even military.I think we must once again accept that colors change enormously over the years, exposure to the sun and the moon. See this Uhlan Rgt.3, 11 or 15 cap. It should be Darkblue.
I'm inclined to agree with Tony that UR18 is the most logical candidate, assuming this cap is even military. Many of the post-1915 "peacetime" Feldgrau uniforms for the cavalry reverted to the old "Dunkelblau" color schemes for their caps - maybe this is one of those? The Feldgrau visor is definitely post-1915. And cockades are easily replaced over the 100+ lifespan of these things. The color, to me, looks legitimately purple and not a chemical or sun fade. You found a weird one, for sure!Nothing to do with Clergy.
What is interesting, UR18 was the only cavalry Regt to change the Regimental colour during the war, adopting purple on 21 September 1915.
But this cap is in the pre-war Dunkelblau pattern with white top. Very interesting.