Hi, emm12345:
Sorry to hear about the damage to your vehicle. It's upsetting to be faced with something so expensive, as well as inconvenient, to be without transportation.
First of all, I'm not a mechanic or a lawyer, but I will give you what advise I can.
If CT (or anyone whom you hired to work on your vehicle) does not perform the work correctly and damage is caused, either during the repair or later as a result of the repair, then absolutely they should be liable for the cost of their mistake.
However, your task now will be proving that it was CT who did the work incorrectly, and then compelling them to pay.
You could gather what evidence you can, take it to them, and attempt to get a settlement. They might do this voluntarily if you have some convincing evidence and they think they would lose a court case. But by the sounds of it, they are currently just looking for any excuse to give you, in the hope that you will believe them and go away.
Based on what you've said, the problem was not with some minor crack in the metal, on a protected part of the engine. Besides, is there even a crack, or did CT make that up? And wouldn't it have only leaked slowly? Instead, it sounds like the filter was installed incorrectly, as your mechanic said. Unfortunately, you may have to got to court and prove this to a judge, who can force CT to pay up. Note that each store is really a separate business, with a unique name and owner, whom you would sue if you go that route.
You already have your mechanic who inspected the problem, and can hopefully give a sworn statement in writing, but may have to testify in court. CT could try other tactics, like claiming they never did any work on your car, ever. They could also claim that somebody else worked on it after them, which caused the damage. I suppose they might also say that it had the crack or damaged filter already when you brought it to them, and that they warned you about it, but you ignored it. Those are all standard defences, so you should be ready to disprove them.
If this case starts to get complicated, you might consider professional automotive and/or legal advice. If you have a CAA membership, you can consult them, too. There is also the Automotive Protection Association, who support consumers.
You might also try searching the CANLII.ca web site for similar cases. There are also a couple of stories on this web site, about suing CT. You can also try contacting the Consumer's Ministry in your province.
Good luck with your vehicle problems - I hope you get the compensation you are entitled to.