None of the above happens with a bonus action shove. It will always be only once per turn, except for I suppose rogues maybe with their additional BA...?
To complete : You forgot to mention that as a BA, it will only be once / turn
but at every single turns because most classes and ennemies won't ever have anything better to do with their BA (than using shove to push, to disengage, to awake a sleeping ally,...)
There's no choice between "shoving or doing nothing" ?
