His encouragement of Louisa's behavior certainly had very little to do with Louisa herself, but was a reaction to his history with Anne. He is certainly resentful, but I'm not sure he is consciously using Louisa to get back at Anne. Unconsciously, yes, perhaps.

But when he talks of his own weakness, I don't think he is talking of what was said or done at Uppercross. I don't think he has realized this. When he says I had been too deeply concerned in the mischief to be soon at peace. It had been my doing, solely mine, it sounds to me as though he is talking only of Lyme: he should have refused to let her jump from the steps.

The connection between obstinacy and weakness here is direct. If by weak he is referring to his entire behavior towards Louisa at Uppercross as well as Lyme, that would imply that he understands his behavior to be a result of his feeling for Anne (because there is no particular reason to regret Louisa jumping from country stiles). He was certainly resentful towards Anne, but I don't think we can blame Louisa's obstinacy on FW's weakness with regard to Anne. Louisa was obstinate because FW would not tell her 'no'-- but he is not encouraging her specifically, intentionally, to hurt Anne.