If you parse this as a ditransitive verb (some sources call it bitransitive), her cat is the direct object, the thing that is being told, and to leave is the indirect object, what the cat was told. That is how this would generally be parsed in modern English, which is (I think) because it would be an incomplete statement without the indirect object. That doesn't mean the indirect is grammatically required, but it is semantically required, so must be provided by context or implication.
Now, you could parse it as a transitive verb with only one object. In that case, to leave would be an adverbial of purpose. That wouldn't make sense here, but in the similarly-parsed "I put my coat on to go out", you can see the meaning of that structure.
Finally, I'm not sure about parsing it as an objective complement. My gut feeling there is that you would phrase it slightly different in that case, making the words of the objective complement be what you might actually say. That parse would make more sense in a case like "I told him no", where the "no" might actually be put in quotation marks or otherwise delineated as reported speech. In that case, I'm not sure if the 'correct' parse would be the ditransitive or the objective complement. The objective complement parse would be more appropriate in verbs that can be used without any indirect object - "I painted the fence" does not immediately raise the question of the colour it was painted, but you can say "I painted the fence red".