When we calculate the relativistic angular momentum of a particle in the direction of the $z$-axis, what relativistic mass should we use? When we calculate the relativistic angular momentum of a particle in the direction of the $z$-axis, what relativistic mass should we use? My hypothesis is that the relativistic mass used for such a calculation does not depend on the particle’s speed in the $z$-direction; rather the  relativistic mass used for such a calculation should depend on the speed of the particle on the $xy$-plane, i.e., the Lorentz factor should only use the square of the speed on the $xy$-plane.  Is my hypothesis correct? 
Relativistic angular momentum in a related discussion suggests that the component of the relativistic angular momentum along the direction of motion between two inertial frames is the same for both frames. It can be inferred from this that the relativistic mass used for calculating the relativistic angular momentum in that direction is independent of the momentum in that direction. Your advice will be much appreciated.
 A: Use relativistic mass.
Remember that adding non-spinning mass to spinning object slows down the spinning of the object. 
I'll give some examples. They all involve loss of mass and increase of spinning rate, instead of gain of mass and decrease of spinning rate. It's just easier to come up with the former kind of examples.
For example if a we have a spinning water tank that leaks water out of its pole, when that tank has lost half of its mass, then its spinning rate has doubled. 
If a truck moving at speed 0.87c carries a flywheel, stopping the truck causes the flywheel lose half of its relativistic mass and its spinning rate doubles. 
If a very hot spinning wheel radiates half of its mass away, its spinning rate stays the same. If we paint that wheel so that it radiates only from its poles, then its spinning rate doubles when its total mass halves. (In the latter case the radiation has no angular momentum)
The reader is encouraged to ponder the similarities between the flywheel in the braking truck and the spinning wheel cooling in space. (One thing that is same in both cases is that mass decreases while angular momentum stays the same)  
Here is an example for people who dislike relativistic mass:
There is a fleet of two trucks, both of which carry a spinning flywheel. One truck starts accelerating towards south, other one towards north. The part of this system called "fuel tanks" loses rest mass. All other parts of the fleet gain rest mass, including the part named "two flywheels". The angular momentum of "two flywheels" is proportional to its rest mass and its angular velocity, the angular momentum of "two flywheels" stays constant, its rest mass increases, so its angular velocity must decrease.
A: A particle is sitting still in an Ecuadorian lab. It has orbital angular momentum because the earth is spinning. Also in this lab there is a linear particle accelerator, it runs from north to south.
When said particle is accelerated in said accelerator, first the E-field in the accelerator has orbital angular momentum, because the earth is spinning, after the particle has been accelerated to high speed, the E-field has lost some orbital angular momentum, and the particle has gained some orbital angular momentum. 
Here's the mathematical part:
The orbital angular momentum of said particle is proportional to its west-east momentum, which is proportional to its relativistic mass. Or it's proportional to the gamma factor, if that's better.  
