I'm sorry for the ordeal you have been through with your cat. It's tough to see a pet you loved suffer problems and pass away. Unfortunately, however, just because the cat died while in the sitter's care does not automatically make either the sitter or Rover liable for the loss of your pet.
There are two areas of law involved here. The first is contract law. You had a contract with Rover. In that contract it expressly states that it is not responsible for any negligent acts of the pet sitter and has no control over the actions of the pet sitter. It also states that you use the Rover service and sitters it refers you to at your own risk. It further states that it's liability to you is limited by the Rover guarantee. That guarantee pays for certain costs you incur while the cat is in the sitter's care regardless of fault, including vet bills, etc. It does not appear to cover burial/cremation costs for a pet that dies in the sitter's care, though. So it appears that Rover is not obligated by contract to pay for it. Under contract law, the company is only obligated to do for you what the contract says it must do.
You also had a contract with the sitter, which I have not seen. Presumably, however, an express or implied term of the contract would be the expectation that the sitter would at least use reasonable care in caring for your cat. But it likely did not provide a guarantee the cat would be returned to you in the same condition that the cat was when you dropped the cat off. However the details of the contract would be important. Depending on the terms of the contract and what happened, you might have a contract claim against the sitter. It may require that you prove that the sitter did not provide the level of care called for in the contract to succeed, though.
The second area of law is negligence law. Under the law of negligence the pet sitter has a duty to you to take reasonable care of the cat to prevent foreseeable harm to the cat. Cats can, of course, get ill, injured, or die even when cared for properly (as can humans and any other living thing). So to make your case for negligence, you would have to prove that the sitter did not provide a reasonable level of care to prevent harm to your cat and that the sitter's failure was the cause of the cat's death. That will be very hard or even impossible to do without knowing why the cat died. Was it something that was out of the sitter's control? Or was it something that occurred due to improper care of the cat? You're the one that has to prove that the sitter was negligent in caring for the cat, so you need to have some facts that show what caused the cat to die and why that was the fault of the sitter. If you didn't have a post mortem done on the cat to determine why the cat died that's a problem because then you may not be able to prove what the cause of death was. And then you still have to tie that to some fault in care provided by the sitter. It would likely require discovery in litigation to get at the care the sitter provided, and that can be a bit costly. If you need discovery that likely means that you can't use small claims court.
Under negligence law, Rover is not responsible for the acts of the sitter unless the sitter was the employee or dependent agent of the company. For that, Rover would have to exercise a fair degree of control over how the sitter does the work. If the sitter is, as Rover claims, an independent contractor then Rover is not liable for the sitter's negligence. However, if Rover knew or should have known that the sitter was a risk to provide poor care (e.g. it had been notified of previous problems with this sitter) it might be liable for its own negligence in referring you to her. Again, discovery is likely needed to get at any evidence that it knew of problems with the sitter. Even if you can prove that, there is the issue of whether the company's limitation of liability in the contract would cut off your negligence claim.
So the bottom line is that you might have a claim against the sitter under either contract law, negligence law, or both, but more information would be needed to nail that done, and the cost to sue and do discovery to get that information is likely to cost you significantly more than $189 if you can't use small claims court. And without knowing why the cat died, it may be impossible to prove your case. Rover appears not liable to you under the contract and might not be liable under negligence law either, unless you can prove Rover knew of problems with this particular sitter or that the sitter was an employee/dependent agent, and either way you still have to prove the sitter's negligence. You cannot win anything for the emotional distress that you and you family have had over the cats death.
But you may, of course, see a personal injury lawyer about it. Most give free initial consultations. That may help you to settle in your mind what you may do and what you want to do.