Style advice
From dictionary.com:
When to Use Whoever or Whomever
Whoever and whomever follow the he or him rule that also works with
who and whom. This trick relies on the fact that him, whom, and
whomever are all spelled with an M. If you can rephrase the sentence
or respond to the question with him, you should use whom or whomever.
You should use who or whoever if the sentence requires he.
Let's try that with "I shall challenge who(m)ever approved of the decision.":
I shall challenge who(m)ever approved of the decision.
I shall challenge him who approved of the decision. (maybe?)
I shall challenge he who approved of the decision. (maybe?)
Still hard to choose, so let's reorder a little bit:
Who(m)ever approved of the decision, I shall challenge.
Him who approved of the decision, I shall challenge. (clearly incorrect)
He who approved of the decision, I shall challenge. (correct)
Restoring the original order gives:
I shall challenge he who approved of the decision.
That means that the original sentence must use "whoever" and not "whomever":
I shall challenge whoever approved of the decision. (correct)
I shall challenge whomever approved of the decision. (incorrect)
But is this reordering valid? Well, let's try it on a simpler sentence:
I shall challenge him. (correct)
Him I shall challenge. (still correct)
Alternatively:
I shall challenge he. (incorrect)
He I shall challenge. (still incorrect)
So this simple reordering seems to preserve correctness and so can be used as I did above.
Sentence that changes meaning
Another way to approach this problem would be to construct a sentence which changes its meaning depending on which of whoever/whomever is used. What I have come up with is:
I shall challenge who(m)ever assured of victory.
So let's examine its two meanings. First the already familiar:
I shall challenge whoever assured of victory.
Whoever assured of victory I shall challenge.
I shall challenge the liar that assured of victory.
and the other is:
I shall challenge whomever assured of victory.
Assured of victory, I shall challenge whomever.
Being assured of my victory, I can challenge anybody without a second thought.
This demonstrates that "whoever" and "whomever" cannot be used interchangeably in such sentences, and that where only one of these sentences is meaningful, the other must thus be wrong.