Perhaps they should be taxed on their MSRP and not be given a tax rebate on discounts. That would encourage them to keep their published prices matched to what they actually charge.
The problem is that as there's no mandatory insurance in the US like there is in Germany, there is a very high chance that the patient may not be able to pay the bill, and so the hospital (or practice) is stuck with the costs. Which may go into six, sometimes seven digit sums. And this loss has to be eaten somewhere... in the form of everyone else having to pay way overinflated sums.
The only way to fix the US health system is to provide a national mandatory insurance for everyone which covers everything of basic healthcare, and then for the private industry they can offer insurance packages for higher-quality care (single rooms instead of shared ones, for example). Or, if that is not possible due to the insurance companies having bought off half of Congress, at the very least introduce a nationwide pricing list for medical procedures, with as few possibilities for deviation as it can be done (for example, services in the center of Manhattan can be more expensive than somewhere out in the woods because rent and staff are more expensive).
This already exists with Medicare.
mandatory insurance in the US like there is in Germany
As for refugees specifically, I'd expect that to be mostly a question of whom they sign on with first. The public insurance has the same rate regardless of provider, so it wouldn't make much of a difference, and governmental support will probably pay that bill anyway.
They get treated first, payment is worked out later.