20 years late to the directors cut ask game but:
Ladybug gritted her teeth, opting to ignore Chloe's taunt entirely. She was instead completely focused on trying to free her arm from Chat Blanc's bite. He'd already set his jaw, blood pooling around his mouth and chin. Like a predator that had captured its prey, Chat wasn't going to let go easily. His claws scraped against the metal of her armour with an awful rasp as he tried to force her back further, to use his own weight to pin her into the dirt. But if Chat was going to be brutally stubborn, then Ladybug could be too.
With as much brute force as she could muster, she forced her trapped hand further into his bite. Her action must have surprised him. He snarled and gnashed his teeth while Ladybug continued to shove her hand against his mouth and jaw. Her blood was spilling down his lips, with each huffing, growling breath from his flared nostrils splattering more red droplets between them. Ladybug gritted her teeth, enduring the searing pain as she refused to let up.
And then Chat Blanc started to gag. But he still refused to release his bite.
It seemed he would sooner suffocate than risk her escape.
So.... You ever seen those attack dogs? The ones that will lunge at a person, bite into the person's arm and lock their jaw there on command? And hold until their owner/master/trainer tells them to release?
That's Chat Blanc. He's been conditioned to attack. Hawk Moth's main target for that was of course Ladybug, and reiterated it time and time again in "training".
But Hawk Moth was also very cautious about what he did with Chat Blanc. He could have told Chat to hunt Ladybug down, but he didn't. Because he didn't train Chat as his hunting dog cat. Chat was trained to fight and destroy.
We see a glimpse here of Chat's fighting style. It's reckless and animalistic. Protecting himself just doesn't fit into it. It doesn't matter how much he bleeds as long as he gets his teeth and claws dug in and holds them there. He's a snap trap. As soon as anyone is close enough he will latch on, he will keep fighting, and not release until he's either broken to pieces - or the one who set him pulls his release.
On the other hand, Ladybug is more coordinated. She is traditionally trained in swordsmanship and fights with a shield at her side. She is adaptive, switching between offensive and defensive depending on the situation (very different from Chat's all or nothing approach).
Ladybug is also (narrow-mindedly) fearless though. She can't resist a challenge if she believes it's within her ability. And she's stubborn. Especially in the public eye.
Fighting is a bit of a performance for her, because she has to meet the expectations of others. She's not necessarily fighting at someone's command or control, but she is held to a role and standard she doesn't dare break.
Ladybug doesn't realise it, but there are a lot of similarities between her and Chat beneath the surface. It's why she meets him head on and isn't scared away by his all-in attack. Instead she's thrilled by it because this is someone else who is giving their all.
Ladybug's just developed her skill into that of a protector, while Chat Blanc's skill has been formed into that of a weapon. A sword and a shield.