Is there a way in the Arcade Labeling profile to access the objects related to a feature?
That is possible. If the idea is to present the user with the attributes of the clicked feature and the related data without the need to click to open the related table, you can do this with the new Arcade functions. If you can explain (show) more about the data schema you have I can help you create an expression.
I have branched your question to a new thread here: https://community.esri.com/message/829126-get-1m-related-objects-using-arcade
Hi Theodore Rakel, Jeffrey Owen, and Lindsy Hales Bentley
I have come up with a solution to symbolizing from a related table in my post:
Symbolizing Feature Class using related table values
https://community.esri.com/thread/215046-symbolizing-feature-class-using-related-table-values
There is a 17 page word document with details on how to accomplish this.