Hi @ReginaC , I think you would have a hard time finding an owner who would go for this as many dogs have not lived with cats and would view them as prey and cats often have a hard time sharing their space with a new cat. Still there may be some cases where it could work, especially a sit with no pets or if you could visit beforehand with your cat to see how s/he gets along with the owner’s pet(s). You would probably be better off joining as a combined member and finding a sitter for your cat while you sat elsewhere.
Here is a link where bringing your own pet is discussed. Cats are mentioned but it’s mostly about bringing your own dog on sits.
https://forum.trustedhousesitters.com/t/is-bringing-your-own-pet-on-a-sit-allowed/