Weird solution in Maya which I came up with: My assumption about transferring transform keys from the whole object to the root and pelvis bones seem to be correct. Though I couldn't make it work just like that. Instead I did the following steps:

1. I installed "Terribilis's" "Mixamo converter" and took the default fbx mannequin from there.
2. Uploaded the manequin to Mixamo and downloaded needed "no skin" animation asset.
3. Converted it with "Mixamo converter" obtaining the reference asset (AssetR). (This asset works just fine in UE4 but it doesn't fit my other skeletal mesh character and deforms it).
4. In Maya I took the initial asset (AssetA) (which is the animation that doesn't work correctly in UE4) , reparented root bone into nothing, deleted the wrapper object.
5. Copied the animation transform keys from AssetR's root and pelvis bones and pasted them into AssetA's root and pelvis bones respectively.
6. Made sure "joint orientation attributes" (which are NOT the same thing as the transforms you casually see on the right side in Maya) of AssetA's root and pelvis bones matched AssetR's root and pelvis bones respectively.
7. Exported edited AssetA's "armature". And now it works fine in UE4.

I would be glad for a clearer solution in Blender. (I can't export Blender's fbx animation into UE4 without my skeletal mesh being deformed - some tips are much appreciated)