For a quick and dirty solution, the justboil.me image uploader/inserter worked in older versions of the code; I haven't tried to see about the current release. You can find it here: https://github.com/vikdiesel/justboil.me. It is not being maintained but if it works then it is a quick way to insert images. You can set the config for limits on file types and sizes and have it resize larger images. One thing I did with the code is that if someone uploads an image with the exact file name to overwrite the existing one. This keeps from having a thousand smileys of the same image from a member; if they like to include say, "big-hugs.gif" smiley in their messages it won't keep uploading it over and over and over; it will replace the existing file. It is easy to also integrate into Dolphin with a bit of php code. What I did was to grab the member's id from the cookie, get the member's nickname (the site used nicknames instead of full names) from the database, then create a directory on the server for them to store their files; each member gets their own directory. I also tied this into the file manager on the site. One thing I want to look at is the responsive filemanager for using as a full blown file manager instead of Moxie Manager which Boonex added to their Dolphin license (I think that has expired as the current Moxie offering is from 2016).
I need to add that this is a TinyMCE plugin. You will upload it to the plugins directory of TinyMCE and have to edit the ChBaseEditorTinyMCE.php to add the plugin and button to the toolbar. I do custom work if you can't do this on your own.
One more edit. This will also allow insertions of images in any area of the site that uses the same TinyMCE init that the messenger uses. To avoid this, you would create a new TinyMCE init and load it into messenger instead of the default TinyMCE init.