I'll sperg some based on my knowledge of the legal system here in leaf land.
Cops certainly lie, but so does keffals. She made it sound like she had an open shut case until the LPS issued their updated statement. Now the only thing she could realistically file a complaint for at the Ontario Human Rights Commission, which supersedes the federal one because it's an Ontario police matter, would be distress from being misgendered/deadnamed due to their system not being updated.
The most pertinent case law paralleling this one would be
Dawson v Vancouver Police Board where a trans woman sought damages against them for purposefully misgendering and deadnaming her while in jail overnight, as well as denying her medical care since she'd undergone genital surgery 11 days earlier. The BC Human Rights Tribunal awarded her $15,000.
In this case they came looking for Lucas Roberts because that's the name their system had, either from some past police encounter we aren't aware of, or from when she (allegedly) charged her ex with rape. They didn't know her new name so it isn't their fault they filed all the paperwork as Lucas. IF her story is true, they misgendered/deadnamed her there, and maybe during booking, based on that. IF the police statement is true, they have camera footage with audio indicating she was not misgendered/deadnamed while in holding. Cameras don't lie. The evidence bags and tags with incorrect name all originate from the booking system having the wrong name.
The police are going to defend themselves by blaming everything on the outdated or insensitive booking system, claim they're updating it, claim they're providing more sensitivity training to officers. Unless keffals can somehow prove they did worse than either she claims or they admit to, she isn't going to get much if any reward at the OHRC. She can't file a lawsuit directly against the police for violating her Charter rights because they didn't profile her or initiate the investigation without cause. They responded to the potential threat the same way they would have against anyone else because THEY thought she was Lucas Roberts, Cisgender White Man With A Gun before they got there. Unless a specific officer somehow brutalized her, she can't get damages against them there either, and it's too late for her to claim that.
TLDR: The whole thing is just a massive grift and she hasn't got a case. Even if she spends the money suing them it'll be like lighting it on fire.