Ok, I found a solution.
Based on the work Zachary Waldowski created (see my original post), I was able to do everything in CALayer.
In a nutshell, I draw one outer circle in the dark, the smaller circle is light gray, the stroked path in white, then draw a triangle for the arrowhead manually.
Here is the relevant section of the code that does the magic:
- (void)drawInContext:(CGContextRef)context {
CGRect circleRect = CGRectInset(self.bounds, 1, 1);
CGFloat startAngle = -M_PI / 2;
CGFloat endAngle = self.progress * 2 * M_PI + startAngle;
CGColorRef outerPieColor = [[UIColor colorWithRed: 137.0 / 255.0 green: 12.0 / 255.0 blue: 88.0 / 255.0 alpha: 1.0] CGColor];
CGColorRef innerPieColor = [[UIColor colorWithRed: 235.0 / 255.0 green: 214.0 / 255.0 blue: 227.0 / 255.0 alpha: 1.0] CGColor];
CGColorRef arrowColor = [[UIColor whiteColor] CGColor];
CGFloat outerRadius = CGRectGetMidX(circleRect);
CGPoint center = CGPointMake(CGRectGetMidX(circleRect), CGRectGetMidY(circleRect));
CGContextSetFillColorWithColor(context, outerPieColor);
CGContextMoveToPoint(context, center.x, center.y);
CGContextAddArc(context, center.x, center.y, outerRadius, startAngle, endAngle, 0);
CGContextClosePath(context);
CGContextFillPath(context);
CGFloat innerRadius = CGRectGetMidX(circleRect) * 0.45;
CGContextSetFillColorWithColor(context, innerPieColor);
CGContextMoveToPoint(context, center.x, center.y);
CGContextAddArc(context, center.x, center.y, innerRadius, startAngle, endAngle, 0);
CGContextClosePath(context);
CGContextFillPath(context);
CGFloat lineRadius = CGRectGetMidX(circleRect) * 0.72;
CGFloat arrowWidth = 0.35;
CGContextSetStrokeColorWithColor(context, arrowColor);
CGContextSetFillColorWithColor(context, arrowColor);
CGMutablePathRef path = CGPathCreateMutable();
CGContextSetLineWidth(context, 16);
CGFloat lineEndAngle = ((endAngle - startAngle) >= arrowWidth) ? endAngle - arrowWidth : endAngle;
CGPathAddArc(path, NULL, center.x, center.y, lineRadius, startAngle, lineEndAngle, 0);
CGContextAddPath(context, path);
CGContextStrokePath(context);
CGFloat arrowStartAngle = lineEndAngle - 0.01;
CGFloat arrowOuterRadius = CGRectGetMidX(circleRect) * 0.90;
CGFloat arrowInnerRadius = CGRectGetMidX(circleRect) * 0.54;
CGFloat arrowX = center.x + (arrowOuterRadius * cosf(arrowStartAngle));
CGFloat arrowY = center.y + (arrowOuterRadius * sinf(arrowStartAngle));
CGContextMoveToPoint (context, arrowX, arrowY);
arrowX = center.x + (arrowInnerRadius * cosf(arrowStartAngle));
arrowY = center.y + (arrowInnerRadius * sinf(arrowStartAngle));
CGContextAddLineToPoint(context, arrowX, arrowY);
arrowX = center.x + (lineRadius * cosf(endAngle));
arrowY = center.y + (lineRadius * sinf(endAngle));
CGContextAddLineToPoint(context, arrowX, arrowY);
CGContextClosePath(context);
CGContextFillPath(context);
[super drawInContext: context];
}