As I understand it, you want the second primary to be wound onto the mandrel so that, when it is later wound on top of the secondary, there is no splice between the two primaries. I can recognize the apparent simplicity of doing it that way.
But, I can also see the complexity of doing it that way in production using an automatic winding system.
The coil winding machine will need two, independently controllable spindles, or they will have to remove the bobbin when winding onto the mandrel.
When winding the secondary, the mandrel will fly around or be anchored in some way to the bobbin. Remember, it cannot be held stationary or the wire from the 1st primary will be twisted to the point of breakage.
Now, to wind the second primary, the mandrel needs to be held but still leave it so that it can spin while the wire is spooled of from it.
I have skipped over the intermediary steps of winding your margin and layer tapes, but those will need to be done and will be more complex with the mandrel connected during parts of the total process.
I'm sure that what you want to do is possible, but may not be practical on standard automatic coil winding equipment without extensive manual intervention.