Inque, as we see her in the cartoon, is different from most of Terry's other foes. Most of his enemies are people with petty grudges or motivations. Feeling that they are owed more than they've received. Spellbinder is like that. Blight and Shriek as well. Then there are the idealists, the ones claiming to have some higher motivation. It may just be an excuse, but Mad Stan at least appears to have an ideology behind blowing things up. Kobra would fall in that category, as would Stalker, who doesn't go after Batman out of a grudge or for money, but because he wants a challenge.
Inque, though, is a professional. Her goal is money, but unlike Spellbinder, she's not trying to steal it because she feels unappreciated. You hire her, she does the job, you pay her, or else. She never seems to be fighting Batman because she's angry about a previous defeat. She's committing a crime, he finds her, they fight. Of his other foes, I think only Curare, a highly trained assassin, matches that approach. It's not personal. Batman is either a paycheck, or an obstacle, but not some scourge of their nightmares.
Inque tries to destroy Foxteca because Powers pays her. She fights Batman because he tries to stop her. She stows away on his Batmobile not out of some desire to unmask Batman and destroy his career because she hates him, but because it's valuable information.
When she escapes from custody with the help of the orderly that's fallen for her, she does capture Batman and use him to lure "the old man" into a trap. That could be viewed as revenge, because it was Bruce using the resources of the Bat-Cave to help Terry that beat her the time before. It could also be Inque knows he's out there and figures it's better to make him come to her at a time and place of her choosing, rather than give him a chance for an ambush. Batman's clearly not going to leave her be, and so the old man likely won't either. Better to just kill them and remove the obstacle.
(Curare's second appearance is similar in that she returns to Gotham hunting the last surviving head of the League of Assassins who put a price on her for failing a mission on her first encounter with Terry. It could be seen as revenge, but since they won't stop hunting her as long as they're alive, from Curare's perspective, it's a necessary step.)
Inque choosing not to kill Terry first, leaving him able to help Bruce when his heart gives out, does tip it into the revenge category, rather than simple pragmatism. But Ingue could argue Terry works best as live bait, and well, professional or no, Inque is more than a bit of a sadist. She enjoys toying with the dumb orderly, and she's fine with killing people by pouring herself down their throats until they choke or burst, or she tears her way out. Even though she could just cut them or strangle them quickly and efficiently.
Also, if Terry's alive, she might be able to use him as a shield if need be. In one of her last appearances in the series, she tries taking a hostage to make Batman back off. It's her poor luck she chose a disguised Superman.
Inque's one soft spot is her daughter, who she sends money, but rarely if ever interacted with until she needed help. It contrasts with Terry, who tries to maintain a presence in his mother and little brother's lives, even as being Batman pulls him away. He wants to protect them, but he doesn't want to do it like Bruce Wayne, keeping them at a distance, trying to control their lives. So even when they don't know what's causing him to be tired, or miss school, they still worry about him. As opposed to Bruce, who has basically alienated everyone who gave a shit about him by being a complete dick. Inque's daughter, who has only known her money through cash gifts, sees her mother's need for help as an opportunity to make some money. That's what she's been taught to value, and she's learned it well.
The brief Batman Beyond ongoing Adam Beechen wrote in 2011 established Inque had immigrated to Gotham from a country torn apart by civil war, and was then sold into slavery, sexually abused, escaped, only to later be experimented on while she was pregnant. Maybe that explains most of it. Not only treating people as a commodity, worth only what they can get you. But that if you have power, you take what you want. If someone tries to stop you, make sure they don't live to do it again.
Terry and Inque are both fight on behalf of others. Terry for Bruce, Inque for Derek Powers and his ilk. Terry takes to Bruce's mission quickly enough. Inque's more of a professional about it, because it's usually just business. She doesn't care why the person who hired her did so, just so long as they pay when she's done. Terry's been on the receiving end of people abusing their power, and decided to try and save others from a similar fate. Inque decided using her power for her own ends was the best way to go. If you can't beat 'em, get hired by 'em.
Inque, at a certain point, chose to be a weapon. Looked at the hand she'd been dealt, and decided she'd do what she was hired for. There's a freedom in that. She can choose who she works for, but she can also tell herself that whatever she does, it's not her. Someone would have been hired, if not her. Terry's more constricted. He's Batman, and that means working for Bruce Wayne, and doing things Bruce Wayne's way. But, for the most part, their morals align, so it works. Terry may not always agree, but their goals are the same.
But given how well he does as Batman with relatively little training, it wouldn't be too hard to see a path where Terry ends up a weapon like Inque. Maybe not a freelancer; more some bigshot's legbreaker. Someone that tells him the anger's fine, use it on the people I point to and you can do what you like. And once you're done, I'll pay you for it.
There's also the fact Inque was the subject of what were probably illegal experiments. She's a product of the same sort of ethically questionable work that Terry's father objected to. The same sort of work that got his father killed. So she's the cautionary tale of that. Terry fought a fair number of one-off enemies who were the result of accidents due to cost-cutting measures. Those people mostly lashed out in anger, and he had to try and calm them down, keep them from hurting anyone or themselves.
Inque's what happens when that person embraces the hand they were dealt, what can happen if that kind of work continues, and she's Terry's most dangerous opponent. The one I'm not sure he ever actually beats one-on-one. Bruce's involvement helps him twice, and Superman takes care of her once. Which should make Inque terrifying to Terry. Other heroes occasionally have that villain they can't beat alone. Venom was like that for Spider-Man the first several years of his existence. Bane took Batman apart the first time they clashed. But both of those guys were fixated on beating the hero. Venom wanted to kill Spider-Man for ruining their lives, and Bane wanted to test and destroy the Bat. Inque doesn't really give a shit about Batman, he's just in the way, or someone she's hired to kill. Makes you wonder what she would be like if it ever became personal to her.
