I am unclear about what happens when I mix those two color spaces in one app. Will the sRGB colors just look paler than without mixing them with the colors of the larger Display P3 color space? Should I be trying to avoid it?
If you are developing for iOS, and color is important to your app, you want to support both sRGB (as it is the standard) and also support P3. You may or may not need separate P3/sRGB assets. The system's color management can adjust P3 colors to display on sRGB, though in some cases you might experience clipping.
To take advantage of color management for your images, you want to use tagged (i.e. embedded profile) images for both sRGB and P3. You want P3 images to be 16 bit per channel, PNG, with the P3 profile embedded.
ALSO, if you are having issues with P3 content not displaying properly on sRGB, then in the asset catalog of your Xcode project, provide separate images and colors for both P3 (wide color) and sRGB devices.