Keep pulling hate off my pet.

What level are you?

In short: Don't start attacking right away. Wait til ~80% to be safe. Once you get hate: Turn Around ! Your pet should grab hate back in a few swings. You may not even want to use Sic because the pet may not do something that will overcome the hate generated by using the pet command itself.

This is part of the reason I've never liked dnc sub for soloing (I'm still 75, just returned the other day). Leveling bst WAY back in the day is how I learned that using items generates hate, resting generates hate, using Fight ja generates hate. Doing anything generates hate. And the lower your level, the harder it is for your pet to hold hate.

If you keep attacking, and worse, healing yourself, the mob will keep attacking you. If you haven't died yet, don't worry about it. If you can tank and live, then go ahead. If you're worried, run out of hate's range before you heal yourself (assuming it doesn't start following you).

But back to my original point:

Turn around. Stop attacking.