Here is what I tried to do to set D3DXPLAY_ONCE flag (not working though):

ID3DXKeyframedAnimationSet* set = 0;
animController->GetAnimationSet(animationIndex, (ID3DXAnimationSet**)&set);
D3DXCreateKeyframedAnimationSet(set->GetName(), set->GetSourceTicksPerSecond(), D3DXPLAY_ONCE, set->GetNumAnimations(), set->GetNumCallbackKeys(), NULL, &set);

// Code here to set the track and play the animation...

From what i can tell D3DXCreateKeyframedAnimationSet only outputs ID3DXKeyframedAnimationSet it does not copy SRT keys from existing animation set, you need to that yourself.
Note that i have not done this before so i am going to guess with help from MSDN documentation:

ID3DXAnimationSet* dieAnimationSet;
ID3DXKeyframedAnimationSet* dieKeyFramedAnimationSet;
 
animController->GetAnimationSet(animationIndex, &dieAnimationSet);
 
// creates "empty" keyframed-animation-set
D3DXCreateKeyframedAnimationSet(
    dieAnimationSet->GetName(), 
    dieAnimationSet->GetSourceTicksPerSecond(), 
    D3DXPLAY_ONCE,
    dieAnimationSet->GetNumAnimations(), 
    dieAnimationSet->GetNumCallbackKeys(), 
    NULL, 
    &dieKeyFramedAnimationSet);
 
// now you need to copy SRT keys data from dieAnimationSet to dieKeyFramedAnimationSet
// some thing here i am not sure how to obtain as documentation is not clear enough
LPD3DXKEY_VECTOR3 scaleKeys[ numKeys ];
LPD3DXKEY_QUATERNION rotKeys[ numKeys ];
LPD3DXKEY_VECTOR3 translKeys[ numKeys ];

for each numKeys and "periodic/local/whatever_time_they_mean_by_this"
{
    D3DXVECTOR3 scale, transl;
    D3DXQUATERNION rot;
    dieAnimationSet->GetSRT( timePos, 0, &scale, &rot, &transl );
    scaleKeys[ i ].Time = timePos;
    scaleKeys[ i ].Value = scale;
    rotKeys[ i ].Time = timePos;
    rotKeys[ i ].Value = rot;
    translKeys[ i ].Time = timePos;
    translKeys[ i ].Value = transl;
}

DWORD newAnimIndex;
dieKeyFramedAnimationSet->RegisterAnimationSRTKeys(
    "animation_name",
    numKeys, numKeys, numKeys,
    scaleKeys, rotKeys, translKeys, &newAnimIndex);

// then here unregister old dieAnimationSet and register new dieKeyFramedAnimationSet to animation controler

Now look how stupid this is for setting such a small and trivial thing.

@belfegor: I'm new to D3DX animation, so I will need a little more help

and "periodic/local/whatever_time_they_mean_by_this"

I don't understand this part

How do I get numKeys?

The following functions doesn't exists:

dieAnimationSet->GetSourceTicksPerSecond()
dieAnimationSet->GetNumCallbackKeys()

I am sorry, i just copied your code from above for that function and didn't look if it is correct.

I commented in code that i don't know for some things how to obtain them.

You could put it like this for test:

D3DXCreateKeyframedAnimationSet(
    dieAnimationSet->GetName(),
    30.0, // for this check in 3ds max and/or exporter options on which value it was set
    D3DXPLAY_ONCE, 
    1, // assume 1 animation in set
    0, // assume no callback keys, otherwise i don't know how to obtain them
    NULL,
    &dieKeyFramedAnimationSet);

Then for numKeys you could try (my guess is that GetPeriod() is in seconds?):

double timeStep = dieAnimationSet->GetPeriod() / 30.0;
DWORD numKeys = std::floor(dieAnimationSet->GetPeriod() * 30);

double timePos = 0.0;
for(DWORD i =0; i < numKeys; ++i, timePos += timeStep)
{
    D3DXVECTOR3 scale, transl;
    D3DXQUATERNION rot;
    dieAnimationSet->GetSRT( timePos, 0, &scale, &rot, &transl );
    ...
}

Maybie there is an easier way but i don't see one.

Now, the animation is not playing anymore :/

Here is the code:

LPD3DXANIMATIONSET tempSet = 0;
LPD3DXKEYFRAMEDANIMATIONSET set;
m_animController->GetAnimationSet(index, &tempSet);


D3DXCreateKeyframedAnimationSet(
    tempSet->GetName(),
    30.0, // for this check in 3ds max and/or exporter options on which value it was set
    D3DXPLAY_ONCE, 
    1, // assume 1 animation in set
    0, // assume no callback keys, otherwise i don't know how to obtain them
    NULL,
    &set);


double timeStep = tempSet->GetPeriod() / 30.0;
DWORD numKeys = std::floor(tempSet->GetPeriod() * 30);


// now you need to copy SRT keys data from dieAnimationSet to dieKeyFramedAnimationSet
// some thing here i am not sure how to obtain as documentation is not clear enough
LPD3DXKEY_VECTOR3 scaleKeys = new D3DXKEY_VECTOR3[numKeys]();
LPD3DXKEY_QUATERNION rotKeys = new D3DXKEY_QUATERNION[numKeys]();
LPD3DXKEY_VECTOR3 translKeys = new D3DXKEY_VECTOR3[numKeys]();


double timePos = 0.0;
for(DWORD i =0; i < numKeys; ++i, timePos += timeStep)
{
    D3DXVECTOR3 scale, transl;
    D3DXQUATERNION rot;
tempSet->GetSRT( timePos, 0, &scale, &rot, &transl );


scaleKeys[ i ].Time = timePos;
    scaleKeys[ i ].Value = scale;
    rotKeys[ i ].Time = timePos;
    rotKeys[ i ].Value = rot;
    translKeys[ i ].Time = timePos;
    translKeys[ i ].Value = transl;
}


DWORD newAnimIndex;
set->RegisterAnimationSRTKeys(
    tempSet->GetName(),
    numKeys, numKeys, numKeys,
    scaleKeys, rotKeys, translKeys, &newAnimIndex);


animController->UnregisterAnimationSet(tempSet);
tempSet->Release();
animController->RegisterAnimationSet(set);

RESOLVED!

I had to simply add 2 lines to play the animation once:

animController->SetTrackPosition(track, 0.0);
animController->KeyTrackEnable(track, FALSE, currentTime + set->GetPeriod());

No need to compress or clone the animation or do anything else

Now to do animation callbacks, is there is any way as simple as this.

@Steve_Segreto: I tried adding this method to my class:

HRESULT CALLBACK Character::HandleCallback(THIS_ UINT track, LPVOID pCallbackData)
{
    MessageBox(NULL, "Callback", NULL, NULL);
    return S_OK;
}

And when updating the animation:

animController->AdvanceTime(elapsedTime, this)

The callback method Character::HandleCallback never fire.