Consider the aphorism "the squeaky wheel gets the grease." Why is this an aphorism? Because squeaky wheels are annoying. If the people who use the wheels have to hear them squeak, they'll soon oil them. So if you hear a squeaky wheel (for example, on a shopping cart), that's usually because the people with the oil don't ever have to get close enough to hear that squeak.
Proposals like Basic Income, or even welfare, basically translate to "greasing all the wheels on a regular schedule, whether you've heard them squeak or not"—i.e., fixing problems that aren't actually bothering you personally. It's nearly impossible to incentivize anyone to do that.
My proposal here is more in terms of "ensuring when manufacturing wheels that they will squeak as loudly as possible when they've begun failing." You want to solve the shopping-cart case, where the person with the oil isn't the person who has to use the cart. And the best way to do that is to ensure that the customers will be too irritated by the squeaking to actually use the carts; and therefore the carts will go disused; and therefore the customers will complain to the cashiers that there are no carts, and people will buy less than they would with a cart, and revenue will go down, and some stakeholder will notice, and gather information, and figure out that it's that all the carts are squeaking horribly, and so oil them.
Or, in other words, you want the squeak of "people not getting treated because they don't have the money" to be loud enough to actually make it all the way up to the hospital's CFO, where the squeaking will cause cost-cutting; and even further, to the city surrounding the hospital, where the squeaking will cause donating.
In fact, if you oil the carts regularly, you might not realize that your wheels are badly-made and are rusting prematurely; or that your parking lot needs a regular dust-blower-ing; or whatever else. Sensitive components—canaries in coal mines, or people with no money in poor health—show you where your system is weakest. If you eliminate them (by just providing UBI, or even just public healthcare), you eliminate the chance to observe where your system fails. In a country with more money than it knows what to do with, this is probably a sensible approach—it has fewer people suffer over the short term, after all, at the expense of a bloated, bureaucratic medical system. But if you're trying to figure out how to cut those costs, you need the squeaking.
Bawawahahaha. No. Non-profit is just a tax status that allows them not to pay income taxes or property taxes on enormous assets and enormous income.
And then, don't allow the for-profit to loan or grant or shift any resources over to the not-for-profit. In fact, ensure they're not owned by the same parent corporation or even the same shareholders.
Keep the logistics consolidated—everything flows through the for-profit's buyer—but then, have the for-profit rent its facilities and machinery, and provide its pre-acquired drugs and materials, to the not-for-profit at market price.
And have the for-profit and not-for-profit hire the same employees and split their shifts; or perhaps, have the for-profit hire them, with the option to volunteer as many hours as they wish to the not-for-profit. (This would lower the expenses of the not-for-profit considerably.)
In other words, leave the not-for-profit "twisting in the wind", where it's still running but nobody's paying for it, so it's just running at more and more of a loss each year. Try to minimize its costs—in fact, have people working for the not-for-profit entirely dedicated to trying to minimize its costs (successful strategies for which can be rolled back into the for-profit.)
But also, treat the not-for-profit as a thing which needs to raise money, rather than just making revenue. A thing like a University, which has entire departments dedicated to getting funding from its alumni. Maybe even work with other not-for-profit hospitals in a Public Advocacy Coalition to build a social norm that if one of these not-for-profit hospitals saved your life, and you went on to great [monetary] success, you should donate to that hospital (and it will, in turn, privilege your family while pretending not to, just like Universities do.)
A fun effect of this is that the not-for-profit will now be a large, visible advocate for cost-minimization in the for-profit's purchasing—since those costs will be passed on to them (so they'll be incentivized to lower them), but they can't just lower them by buying cheaper themselves (because they don't have their own purchasing/facilities management/IT/etc. departments), they'll instead have to figure out how the hospital as a whole can lower costs, in order for them to save any money.
Essentially, this is doing the same thing that creating municipal vote districts along income lines does: the rich voters and the poor voters each get their own voices in how the city should be run, and so the poor voters (through their representative) get the chance to argue against proposals that would benefit the rich voters but hurt them. Right now, hospitals don't have any such representative for their "poor voters."
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(Yes, I know that a lot of hospitals are already not-for-profits. They're still run on a model that tries to maximize revenue, though, because they need that money. If you split the unprofitable patients out, suddenly the for-profit hospital can lower its prices, while the not-for-profit is forced to lower its costs.)