You asked for help. I'm going to be tough and give it to you the best I can. I'll be taking this apart bit by bit to show you your part of things here the best I can. Hang on, this might be a ride!
How am I responding in ways that drive my resentment up?
Your resentment is your feeling. I don't know exactly where it comes from, and even if I did, it wouldn't help you for me to be telling you about it.
Ask yourself some questions: What am I feeling about her? Think about the last few interactions, and what you were feeling immediately after you got a text or had a conversation.
Give yourself some time to just feel these things, you don't have to DO anything at all.
You may answer your own question about what you are doing that is driving up your resentment.
I don't show her the resentment do I?
I'm not watching your interactions, so I don't know.
pwBPD are usually hypersensitive to the emotions of others around them. If you are feeling a tenth of an ounce of resentment while you are talking to her, she will probably notice... .and react.
So tell me what I'm doing wrong... .This is where I start to feel like a failure. It feels like I can't respond correctly no matter how I respond.
It is really tough to do this well. Perhaps the first problem is to define "respond correctly."
If your definition includes the behavior you want from her, you will never get it "right"
If your goal is to reconcile with her, offer her positive, validating interactions with you. Don't validate the invalid. Don't be a human punching bag for emotional abuse. You will still need your boundaries.
I tell her I won't drop something off at her doorstep, and she meets with me only to pick it up and won't sit down to have lunch with me.
You were trying to force her to spend time with you. She saw that as controlling and manipulative, and didn't react well.
If you offer something, offer it as a genuine gift, without attaching strings. If you are feeling too pissy to do this, that's OK. Save the "gift" for later when you can give it freely.
I tell her that I won't discuss it on text and she won't meet with me.
It is your right to not discuss something on text. There are lots of good reasons for that; I'm not second guessing that.
You have no right, no ability, and no good reason to force her to discuss something in your way.
If she reacts by not discussing it at all, or cutting communications for a week, that is her choice, and you cannot stop her... .and absolutely shouldn't try.
I tell her I'm not paying her bills and I'm still the a$$hole and she says that "they will never see us again".
You have every right to tell her what you are going to do with your money.
She has every right to call you any names you stick around and listen to.
She doesn't have to be correct. She doesn't even have to be truthful.
And you don't have to listen to her, or believe her.
It seems that no matter what I do, I can't gain control of this and the ball is always in her court... .
That is true. Here's the kicker:
When you are trying to control her, you are putting the ball in her court, setting her up to smash it right down your throat.
If you don't like how that plays out, stop doing your part of it.
She still has my laptop for God's sake... .
This sounds like the resentment you were asking about earlier.