I had the impression that RoKo underestimated Bubbles because she did not expect the AI in a Human-AI partnership to take the lead, look at most examples we have and you have a fair argument that while AI have agency they're somewhat dependant to some degree on their people. They are all companions in some aspect, even Pintsize.
Bubbles breaks this trend, being quite autonomous and choosing the union with Faye because she wishes to.
On the other hand, you do have Corpse Witch, as an earlier entrepreneurial example, and Punchbot, as a CPA. Roko herself seems fairly independent. Now, of course, there is the fact, that she's only ever met Bubbles before she took her armour off, when Faye was standing up for her, and she was heavily limited by the CW's proverbial stick behind the door.
Part of this view might just be that we're fairly limited in the AI's we've met up to now.
{political digression}
Framing your question in terms of "insurance" is already implicitly limiting it to a commercial transaction and thus subtly sidelining the possibility of fully social healthcare.
{/political digression}
I think it depends on how you frame it exactly. Where I live, there's the state's health insurance, which is run by the state, and a number of not for profit organisations. Partially it's a holdover from earlier union efforts.
As for chassis insurance, I think we need to first solve the issue of who is the legal owner of the chassis in question. It's not that hard to imagine a corporation investing in a certain type of chassis, and putting it to its (AI) workers disposition. Or indeed, a human companion buying one for their AI companion. Add to that that it seems very easy to switch chassis,...
AI are independent people with civic rights, that much we know. On what basis do they have those rights? Is it on the basis of consciousness, i.e. just their mind? If a certain chassis were to be coupled to make up the individual AI citizen, that would be problematic, and more than one of our characters is potentially in trouble.
Insurance efficiency, considering the chassis to be essentially disposable, I should think is much the same as with a car insurance.
Now, another question; how much is Union Robotics going to charge officer Basilisk, if the issue turns out to trace back to a certain raccoon incident?