Hi, Russ,
It is either a broken link or the clip has been somehow been corrupted or deleted from the iMovie Original Media folder. See if you can restore the link by dragging the original clip from the media browser that displays when you click on the project's name under the Project Media item at the top of the iMovie media sidebar. Just drag the clip from the media browser onto the project's name under Project Media. If it is a broken link you would not need to convert it to Mp4, as I doubt that .avi is the problem unless you are running Catalina or Big Sur. If you have the original source .avi file you might try reimporting that one.
The notation on your post shows that you are using Mac OS 10.11, El Capitan. Is the notation outdated or are you in fact using El Capitan? If using El Capitan then .avi would still be supported on your Mac. It would not be supported if you were running Catalina or Big Sur. Since, in your case, other .avi files are being recognized, it is unlikely that your problem is a legacy format compatibility issue. It looks more like the link to the file has been broken or the file has been deleted. When you select the clip and do a File/Reveal in Finder is the clip displayed in the Finder Original Media folder that pops up? If the clip is there then that would indicate a broken link. If the clip is not there then it has been deleted from the library and is gone.
With Mojave, you could do a File/Check Media for Compatibility and then, assuming that the clip is still in the Original Media folder, convert it in the pop up window to preserve its edits. So if you can upgrade to Mojave you 10.14.6 you could try that route. Otherwise, as you mention, if you convert a clip separately the edits will not be preserved.
I don't know of any way of "tricking" iMovie to recognize your Mp4 conversion as the original .avi clip so as to preserve the edits. The Mp4 clip has a different extension and thus iMovie would recognize it as a different clip. So the edits would be gone unless you had a Time Machine backup or other backup where you could export the edited clip and reimport it into your current project.
For an experiment, try looking in the iMovie backups folder to see if you can find a previously saved backup of your iMovie library that contains your current project. To get to the iMovie Backups folder, follow this file path from the Finder menu:
Go/Home/Library/Containers/com.apple.iMovieApp/Data/Library/Caches/iMovieBackups
When you get there you will see a list of previous backups of your iMovie library. Click on one dated just before your issue started. iMovie will open in that library. Navigate to your projects browser (where your projects are displayed as icons) and see if your project is intact and if the .avi file is recognized.
-- Rich