I don't believe that is possible. Lightroom utilizes a catalog, and images must be added to the catalog before it is possible to edit them. Lightroom functions differently from a photo editor such as Photoshop which enables you to browse images wherever they are located. With Lightroom the images must be added to the "system" before they can be seen and edited.
Is there a way to set that up or to add it to the list of editors available?
Lightroom should already be on the list of apps that iOS automatically suggests. In the demo below, I select a photo in the iOS Photos app, tap the standard Share icon, and swipe the suggested apps sideways until I see the Lightroom icon which is already on that list.
Be aware that if you send a photo to the Lightroom app, Lightroom will immediately attempt to upload it to Lightroom Photos in the cloud, and then sync that photo to Lightroom on any other devices you have. In other words, Lightroom is not built to be a purely local image editor.
If you want to force a local-only edit, disable sync in the Lightroom app, do your edit and export, and then delete the photo before enabling sync. You must delete it twice; the second time is from the Deleted album. If you don’t remove it from the Deleted album, Lightroom will upload the deleted image anyway, to give you the option of un-deleting it over the next 60 days before it is permanently removed.
If I'm understanding the question correctly, Digital Rebel is not asking about sharing, but rather editing photos directly. When you choose a photo in the camera roll, and tap "Edit," you go into the iOS photo editor. From there, you can choose additional editors, like the built-in Markup, as well as other apps you may have installed, like Skitch and Camera+ in my case. See the screen shot below.
Unfortunately, Lightroom doesn't fit into this kind of workflow. As noted above, it is a subsystem of its own. You can put photos in and export photos out, but it is not designed to be a "one photo at a time" edit-only application.
Good points from Michael about the difference between using Lightroom as an editing extension, and the sharing method I showed. If you are using external editors on iOS or macOS that can update Apple Photos with the edited version (as an iOS/macOS adjustment “extension”), that won’t work with Lightroom. Lightroom is not set up to send adjustments back to Photos that are connected with the original; in other words it does not work as an extension to Apple Photos.
So using Lightroom to edit from Apple Photos can be practical if you want to export from Lightroom without going back to Apple Photos, but not if you want to bring the Lightroom edits back to Photos. Because Photos and Lightroom are both bulk photo managers, but with different databases and edits.