To explain how classic rigs handle this problem, I assume that you want to export as FBX two animations: one done with FK and one done with IK.

In object mode select the rig and press Shift D to duplicate, then repeat to get three copies of the rig.

For clarity, rename the copies as "IK_rig" and "FK_rig", rename the original as "DEF_rig".

Assign the IK animation to the IK_rig.
Assign the FK animation to the FK_rig, deleting its IK constraints and its IK control bones.

In the DEF_rig delete all IK constraints and IK control bones, and set the parenting with every child bone unconnected to its parent bone.
Then in Pose mode add to every bone two "Copy Transform" bone constraints, targeting the corresponding bone, the upper to the FK_rig, the lower to the IK_rig.

Then choose a bone (in my example the thigh bone), add a custom property, use the gear button to rename it as "FK_IK", right click on its value and choose "Copy as new driver".

Go to the influence value of the lower Copy Transform Bone constraint (that is targeting the IK_rig), right click and choose "Paste Driver", repeat for every bone.

Now press Shift D again to make two new copies of the DEF_Rig: rename them as "IK_Export_rig" and "FK_Export_rig".

Select the IK_Export_rig, check that its custom property slider is set to 1.0: it should play the IK animation; select all bones in pose mode, go to menu > pose > animation > Bake action (clear constraints, visual keying).

This is the rig and the new IK animation you will want to export as FBX.

Repeat for FK_Export_rig, turning its FK_IK slider to 0.0.

[![enter image description here][1]][1]

In classic rigs, all the three rigs (IK, FK, and DEF) are joined into a single rig, but even then you will have to perform the bake function in order to export as fbx, as Blender constraints are not directly exportable.

The FBX exporter has an option to automatically bake all DEF bones, but I made you do it manually to better understand what happens "under the hood".


  [1]: https://i.sstatic.net/pziZC50f.jpg