If you want to go by stereotypes, the average bully cliché is 783.
But everything you highlighted about 3 has nothing to do with empathy. 3 are doers and strive for efficiency. They put their own feelings apart in the name of it (like 5 and 1, same triad).
To make it easier: I want to be admired because I don't have a sense of self. I'll find it in the way other people perceive me. There's no empathy in this.
Your example for 4s is valid but circumstantial and won't result in testing (which was your initial point)
Enneagram 3 - Key Motivations: Want to be affirmed, to distinguish themselves from others,
to have attention, to be admired, and to impress others.
So if enneagram 4 is a feeling type who values individual authenticity and defines their identity internally, enneagram 3 is the exact opposite, a feeler type who do not have internal feelings defining their personal preferences and identity but defines their identity based on how much they are loved and admired in their social group.
Enneagram 4 is therefore Fi-dominant behaviour whereas enneagram 3 is Fe-dominant behaviour. Fe in dominant position measures how objects in the external environment feel toward the individual, and to get a good feedback from others (i.e. love and self-worth), enneagram 3s (Fe-doms) keep adjusting their behaviour and image so that it will garner them the most positive feedback (admiration) in the society, by measuring what is valued the most in their society for their gender. They are also known for being charming and diplomatic.
Therefore, their measure of success is defined by embodying the image of the most praiseworthy individual in their society, according to their society's values rather than their internal values (where the latter would be enneagram 4 behaviour). Their competitiveness emerges when there are rivals in their environment who garner more social praise (popularity) then they do.
Empathy is derived from having an abundance of feeling faculties. People low in feelings do not feel internal remorse for their actions so bad feelings do not serve as an internal deterrent for them but external consequences might.
I understand that my understanding is very subjective but it makes perfect sense to me.