Hi Lost1223 and welcome to the group

Many members here have also arrived feeling exhausted and overwhelmed in their BPD relationships; you're not alone. Adding parenting on top of that and it's no wonder you've got nothing in the tank.
Staying in, and/or improving, a BPD relationship can be pretty unintuitive and take different "moves" than a more "normal" (I use that loosely) relationship. For example, in a "generally normal" relationship, there's a long-term balanced give-and-take. It's expected, whether tacitly or explicitly, that partners will share tasks, whether sharing the same task between them, or each taking ownership of different tasks, or achieving an agreeable split some other way, or over a long time span (i.e. one partner working out of the home for 5 years, then trading to the other partner).
A BPD relationship doesn't usually function according to "generally normal" rules. It can be thought of as an "emotional special needs" relationship. If your partner used a wheelchair, you would have to adjust your expectations of what your partner was really capable of physically. Otherwise, you'd grow resentful over time: "why do I always have to reach the stuff up high, why doesn't she ever have to do that, it isn't fair".
Your wife, if BPD is in play, is similarly impaired... though not in a way that is as visible as using a wheelchair. BPD is a real and serious mental illness that impacts the sufferer's worldview/"lenses", ability to regulate emotions, bandwidth before getting overwhelmed, and ability to have and maintain close relationships, among other areas.
There's a sense in which "what you see is what you get" with her, it seems like. It isn't fair that she did not take care of your youngest when you were ill... and that's the disorder, right there. It is possible that she is a severely limited person and the way it showed up was her not parenting when you needed it.
What's important to remember is that
you get to make choices in all of this. If you choose to stay, we're here to walk with you as you learn and apply
new, counter-intuitive tools and skills to make
your life more livable. The approaches we teach here aren't to "fix" her... they're to help you find a life worth living if you choose to be in a BPD relationship. People stay for all kinds of reasons and we respect that here.
Probably the "worst of all worlds" scenario would be to stay in the relationship without
radically accepting her limitations and without building the key tools of
validation and
boundaries. basically, having a "leaving" mentality while remaining together.
The best possible scenario is you learning, building skills, practicing, and growing, modeling a healthier path forward for her and the kids, and being OK with her choosing whatever she chooses.
While there are no guarantees in a BPD relationship, what comes pretty close is the knowledge that the only person we can control is us.
...
am also wondering if anyone has any suggestions for how to deal with the exhaustion.
My H's kids' mom has many BPD type traits and behaviors. While the kids are older now (16 & 18), when they were younger, it was really difficult to coparent. Trying to get agreement was like pulling teeth and led to resentment and frustration on our part.
What we learned over the years is not to rely on her for stuff we needed to happen. BPD impaired her ability to cooperate, plan, and follow through.
When BPD is in play with parenting, it is much more effective to plan all child care, transportation, support, babysitting, meals, etc, on your own (i.e., deputizing others to help you out -- teenage baby sitter, grandparents, sending the kids to day camp, Walmart meals, school aftercare), versus asking/hoping/expecting a parent wBPD to help you when you're exhausted.
It isn't fair and it isn't how you wanted it to be, but if you can outsource as much kid help as possible, that is both accepting that she has serious limitations, and gives you a break so you aren't at 110% exhaustion all the time.
We do all the transportation for our 16 year old right now. It is 100% us and 0% Mom. It isn't fair or equitable, but it is effective and safe. I no longer resent her "not doing her part" because she wasn't very reliable anyway. If we were getting overwhelmed with driving the kids around, I'd probably ask my parents for help.
This is not normal-range parenting but you need to get yourself well rested and closer to baseline so you can function long term. Once you have sleep/exhaustion more under control, I think a lot of the other pieces will fall into place.
What do you think?