We found that for some custom objects that were tied to existing projections we needed to really jump through hoops to make them visible.  I know it isn’t like-for-like what you’re asking but let me see if I can find some of the notes… it was really very convoluted.  It might help. 

Nick

Here are some notes that might help , but in our case we were deploying to main rather than int:

The custom projection needs to begin with an upper case letter when created.

We deployed into the Users category for main, but I can see from a screenshot that Integration was an available category for us in Apps10.  Perhaps you did not choose this during the creation process?

When you go through the configuration of the custom projection make sure you mark it as Approved 

Once in place obviously be sure to Publish it

The rest of the notes I have are really specific to main rather than int so likely won’t be of any use to you.  You’re a smart guy so I’m sure you’ve spent time on this already, and the above seem rather obvious, but maybe they will help.

Maybe in your case the question is really “how do you use/see the Integration category when creating a custom projection?”

Nick