Here is my best interpretation of the code:
240.15: Current carrying conductors that are ungrounded (example, hot wires in a US AC circuit panel) must have overcurrent protection.
240.22: Current carrying conductors that are grounded (example, neutral wires in a US AC circuit panel) do not need overcurrent protection.
706.31(A): The same rules apply to DC wiring. Any current carrying conductor, negative or positive, must either be connected to ground somewhere, or have overcurrent protection.
If there is a ground connection in a DC system, it's usually the negative. But from what I've read, depending on exactly what make and model equipment is being used, grounding either conductor can mess up ground fault protection, which is also required by code (690.41) if you have a solar input. I'm not 100% sure on this part, but I think its correct. So it seems the best way around this issue is to put both the positive and negative on their own breaker.
If you have more than one battery, it gets a little more complicated. A bank of batteries in parallel should each have an overcurrent device on at least one conductor. I don't think it matters which. The batteries I've seen have the negative controlled by the BMS, while the positive has a breaker built into the battery. (I'm still looking for the code on this. It's got to be in 706 somewhere.)
From there, the best practice seems to be to put all the conductors on a set of buss bars (is it buss or bus?) and have a single set of conductors (one negative, one positive) going to the inverter. Each of the individual battery breakers would be rated for the max current of the individual battery. The big set of cables would then have their own overcurrent devices. One for the positive, one for the negative. These overcurrent devices would be rated for the maximum current of the combined battery bank. Or the maximum current rating of the inverter. Whichever is smaller.
250.4(B): Now, DC devices are still going to need a connection to ground even though the conductors are not grounded. I haven't figured that part out yet, but that wasn't my original question anyway. 250.122?
I hope this all makes sense and is accurate. Please correct me if I'm wrong.
Edit: I'm adding NEC 2020 code references where I can find them.