The Empress spent her entire life coming up through the ranks of a Universe that uses an Agonizer Booth as a daily punishment for ANY Infraction.
Somehow, I can't see what appears to be a similar form of torture in 'Phaser' form, being that big of a hindrance for her.
It had very little to do with "Badass" and much more with being acclimated to that particular form of energy.
I had a chronic pain condition through my 20s and 30s, got it sorted thanks to natural medicine, but it's left me with a pain threshold you only usually see in action movies. I freak out paramedics, nurses, doctors, because you can pretty much sew me up without anesthetic, stick needles in me, whatever you want, I don't flinch, I don't feel pain for something as trivial as that, and because I treated it naturally, my body got better at healing and scratches and bruises just go away. Hot wax tickles

In that environment, with the life she'd have lived, agony booths, a sadistic mother, Georgiou's body would be hardened, tempered and resilient, her pain threshold would be through the roof, and she wouldn't be the least bit bothered by a bloody nose or a bit of phaser burn. She could walk that off.