I solved it using a context menu:

In my view controller:

        let interaction = UIContextMenuInteraction(delegate: self)
        annotationTypeButton.addInteraction(interaction)

And the delegate as an extension:

//MARK: - UIContextMenuInteractionDelegate
extension AnnotationDetailsViewController: UIContextMenuInteractionDelegate {
    func contextMenuInteraction(_ interaction: UIContextMenuInteraction, configurationForMenuAtLocation location: CGPoint) -> UIContextMenuConfiguration? {
        return UIContextMenuConfiguration(identifier: "annotationTypeMenu" as NSCopying, previewProvider: nil) { _ in
            let children: [UIMenuElement] = self.makeAnnotationTypeActions()
            return UIMenu(title: "", children: children)
        }
    }

    func makeAnnotationTypeActions() -> [UIAction] {
        var actions = [UIAction]()
        for type in AnnotationType.allCases {
            actions.append( UIAction(title: type.rawValue, image: type.image, identifier: nil, attributes: []) { _ in
                let annotationType = AnnotationType(rawValue: type.rawValue) ?? AnnotationType.tips
                self.annotation.type = annotationType
                self.configureAnnotationTypeButton(with: annotationType)
            })
        }
        return actions
    }
}

Works nicely, but I do get an autolayout error in the console on opening the menu ...