It's not so much the hitch itself, but what you are pulling.
A dog trailer can't weigh all that much in my mind (I could be wrong). With that assumption, I would think you are safe without upgrading your cooling bits. Having said that, it never hurts to have additional cooling capacity (that's why I made sure to have the factory kit even though I don't expect to tow at the limits of Class II)
In your case, I don't think it would be necessary to add it . BUT If you have cash to blast, by all means, mod that thing!!