How would I go about getting this casing replaced?

Your best bet with this kind of damage is usually to buy a donor laptop with unrepairable damage, in good cosmetic shape in many cases. I've had to do it to my E6440 when I dropped it and broke the midframe since it broke near the VGA port and that affected a LOT more then being a scar - the power jack was loose as well, but undamaged. I ended up getting a stripped unit with a bad motherboard and no CPU to get what I needed, and extra parts as I needed them. You can get these broken laptops for almost nothing because they're not easily repaired and tend to only be good as donor machines. The issue with consumer grade laptops is they are pretty much all built to be replaced with how much major parts cost. Unless you can source a good donor for the parts you need, the cost of parts like this tends to not be worth it.

With MAJOR service parts like palmrest assemblies on their own, the cost tends to be about as much as a junk laptop so you’re usually better off scrapping it or buying a junk machine with the parts you need in the vast majority of cases, unless the bare part is reasonably priced or it’s a battery - never buy a used battery.

For example, this is usually what I buy when I need multiple major parts - especially when I need to share between 2+ systems as the parts cost is no longer viable and it’s otherwise unrepairable as a result I won't feel bad stripping down: