Hang on, I just recalled it's actually possible to auto it, just gotta set up extra steps.
One large breeding room with the salamander , One slave cage with two slaves and Two birthing rooms.
When the pixie auto moves a slave pregnant with the salamander egg quickly drag the other slave to it and it'll reset the pride trigger. Once the first slave gives birth and the other gets pregnant it'll move the second slave to the second birthing room and move the first slave back in.
I only recalled this because when I'm mass moving my family slaves to the site for milking they tend to end up taking a spot in a slave cage that's already occupied and when the pixie wants to move the milked slave back it can't, just says waiting for position until I either; 1. Manually move the milked slave to an empty slot elsewhere or 2. Move the slave to do something else then the pixie can put the first slave back.
I assume it's not confusing but it's the general idea I got