iOS Sort Menu with Few Items/Options

I have a screen for browsing the app's database of leaf species. I am trying to implement a way for the user to choose how they want the database to be sorted for their browsing. When you toggle the search button, a search bar appears on the top of the screen. In the search bar, I've replaced the bookmark icon/function with a sort icon. When you click the sort icon, a picker menu drops down with the three sorting options. I implemented a picker menu because Apple doesn't have a design patter for drop down menus/lists.

This is what it looks like:

I chose not to use an action sheet because those pop up from the bottom, and that kind of experience/response would be confusing triggered from a button press at the top of the screen. I also chose not to use a segmented control (another common sorting pattern)because there is already a segmented control for toggling browsing two different libraries and I though two of them on a single page would be too much.

That being said, I am really dissatisfied with how it looks. I also think that a picker menu is ridiculous/too much here - the UX is bad because it doesn't feel intuitive to have to scroll to select when there are only three options.

Any ideas on how I could implement a sorting feature/menu for this screen? Thank you so much in advance.