Try this:

FormObject[<|"CHOOSE: " -> <|
    "Interpreter" -> {"A" -> 1, "B" -> 2, "C" -> 3, "D" -> 4},
    "Control" -> (RadioButtonBar[#, #2, Appearance -> "Vertical" -> {2, 2}] &)
    |>
  |>]

The control value should be a function with a slot (#) to accept the dynamic local variable created by FormObject and another (#2) to accept the "Interpreter" argument.

The value you see in your output, "~216d9372a6753210", is the same as I see. It is the initial value of the local variable. An odd choice. I'm not sure why, except that it is unlikely to match a valid value for any control. It is followed by the interpreter. Note the two are between brackets [...]. That means your static radio-button bar is called as if it were a function, and two arguments were passed to it. Like this:

RadioButtonBar[
  "A", {"A", "B", "C", "D"}, 
  Appearance -> "Vertical" -> {2, 2}
  ][Dynamic[var], Interpreter[{"A" -> 1, "B" -> 2, "C" -> 3, "D" -> 4}]]

In my suggestion, the call ends up being

RadioButtonBar[
  Dynamic[var], Interpreter[{"A" -> 1, "B" -> 2, "C" -> 3, "D" -> 4}] 
  Appearance -> "Vertical" -> {2, 2}
  ]

(The variable var is localized and created internally. We don't need to know exactly what it is.)

Fiddling around, I discovered this works, too:

FormObject[<|
  "CHOOSE: " -> <|
    "Interpreter" -> {"A" -> 1, "B" -> 2, "C" -> 3, "D" -> 4},
    "Control" -> (RadioButtonBar["A", {"A", "B", "C", "D"}, 
        Appearance -> "Vertical" -> {2, 2}] &)
    |>
  |>]

So it is an alternative to my first suggestion.

Caveat: I haven't used forms, and I don't know which of the two alternatives is to be preferred. Someone else will have to answer that.