Tim can't want Tully evicted because he's put her in a position where she's at the centre of attention.
He had enough foresight to know Sharon and Jasmin were likely evictees, he knows that getting Tully saved is in his best interest. By making the easily manipulated and reactive Tully his enemy, he gets more public support and makes her a part of his "Tim show". Airtime means she is more likely to get votes to save. He wants her there. Getting rid of Caleb or Ed is his top priority. He's just doing it in a way that won't make people suspicious that he wants Tully to stay. Nominating Tully and Caleb together and putting himself up is all part of the plan. He won't want Jade out either because she is part of his plan to ruin Ed's Mr perfect image.
That's what I think Tim's strategy is anyway. He doesn't bother with people he can't manipulate, he just makes plans to get rid of them. People like Heidi or Tully who react to him and think they have him worked out are just giving him what he wants.