I went back to your site to confirm the issue with the source for the featured image, and in setting the template to be assigned to the intermediate post type it broke the template.
I don't know why that happened, because it didn't happen when I first did it, as described in my previous reply.
I've looked in the settings in the database and I can't see what would cause the problem with the broken template; if you were able to fix it before via the code editor, hopefully you can again. Sorry about that.
Rather than risk doing anything else on your site I created a sandbox where we can test this, but I didn't run into any problems, and everything worked as expected (with the proviso that the template must be assigned to the intermediate post type for its fields to be available in dynamic sources).
Here is a link to log in: hidden link
I made a relationship between post types Left and Right, each of which supports Featured Images, and there is a relationship field that stores the sort order.
I created a Content Template ("Intermediates") that outputs a mix of fields from the intermediate post type or one of the related post types.
I created a View directly in the Left post "My Left Foot" to query the intermediate posts, ordered by the ordering field, and inserted the template in the output of this View.
If you familiarise yourself with the set up and then visit the My Left Foot post on the front end you will see that it works, and I don't think you will encounter any problems editing the View or the template.
One observation about your statement: I used it in other places as well before, but that doesn't work now anymore because the context changed.
That is expected. A simpler template that only outputs fields from the "current post" would work in many different contexts, but a template that outputs fields from related posts where it matters what the origin post is can only really be used when displaying posts of that type.
In that sense, there is probably no reason not to assign your template to the intermediate post type (you won't ever be displaying these directly).
The challenge then, is to try and identify why this set up is failing on your site.
Might I suggest that you try editing the sandbox site and make changes to the template comparable to what you have in your own template, and see if anything breaks and take note of what causes it when it happens?