Here are some useful calls I've developed for editing FBX animation files using the Python FBX SDK.

fbx_animation_editing.py

import FbxCommon


def fbx_trim_keys(scene, start, end, take_buffer=10):
    """
    Uses the FBX SDK to remove unnecessary keys.

    :param scene: FBX SDK scene class that hold data to edit
    :param int start: start frame of animation keys to keep
    :param int end: end frame of animation keys to keep
    :param int take_buffer: number of frames to keep as head and tail of clip
    """
    # Get animation stack and associated layer
    stack = scene.GetCurrentAnimationStack()
    layer = stack.GetSrcObject()
    
    # Build FbxTimeSpan class for keys that should be saved
    anim_start = FbxCommon.FbxTime()
    anim_start.SetFrame(start - take_buffer)
    anim_end = FbxCommon.FbxTime()
    anim_end.SetFrame(end + take_buffer)
    anim_range = FbxCommon.FbxTimeSpan(anim_start, anim_end)
    
    # Iterate through all animation nodes in the layer
    for l in range(layer.GetMemberCount()):
        anim_node = layer.GetMember(l)
        # Check node has keys
        if anim_node.IsAnimated():
            # Each node has a curve for every animated attribute
            for c in range(anim_node.GetChannelsCount()):
                anim_curve = anim_node.GetCurve(c)
                anim_curve.KeyModifyBegin()
                # Iterate though every key on a given curve
                for k in reversed(range(anim_curve.KeyGetCount())):
                    key_time = anim_curve.KeyGetTime(k)
                    # Check if key's time is not inside the range to save
                    if not anim_range.IsInside(key_time):
                        anim_curve.KeyRemove(k)

                anim_curve.KeyModifyEnd()


def fbx_name_take(scene, name):
    """
    Maya's FBX plug-in is limited in how it allows you to name take. Using the FBX SDK we can pass a custom string.

    :param scene: FBX SDK scene class that holds take to rename
    :param unicode name: name for animation take
    """
    stack = scene.GetCurrentAnimationStack()
    stack.SetName(name)


# Example
animFilePath = "C:\\Users\\andrewc\\Desktop\\test.fbx"
manager, scene = FbxCommon.InitializeSdkObjects()
result = FbxCommon.LoadScene(manager, scene, animFilePath)

fbx_trim_keys(scene, 30, 60)
fbx_name_take(scene, 'SuperCoolAnim')

FbxCommon.SaveScene(manager, scene, animFilePath)

Do you know in what way it would be possible to use this to actually clip the timeline ranges for the exported file? (as opposed to saving them to 'takes'). More specifically, exporting each take as a separate .fbx file?