This really feels like a bug or design flaw, but it may actually be intentional. Without this restriction, the player may be more prone to avoid the only action that will allow Deacon's story to progress any further. So, this could just be Bethesda's way of "railroading" the player.
The proper solution (per the game's current design) is to actually read the paper containing Deacon's recall code, and then say it to him in conversation. Until you do this, you can work-around the unavailability of a "Dismiss" option by taking on another companion (which forces you to dismiss your current companion). However, there's no non-hostile way to get him out of Power Armor without reading the code.
In fact, the recall code Deacon gives you is fake. This will only be revealed to you when you try reading it to him. At that point, he'll tell you that the code was just a lesson for you to learn. Given that lesson, it's really hard to say what you should believe about Deacon's true nature past this point. It's still actually possible that he is a Synth and the code he gave you just wasn't one that will work.