"...shunt dying patients around like sacks of rice"

Well, that is certainly colourful, but I'm not certain it's fair.  It is entirely reasonable for a faith-based institution to act in accordance with their faith.  Stating that people have rights, not institutions, is disingenuous - people have  the right of association - they have the right to form organizations that adhere to their beliefs.  To insist that an organization go against their beliefs is to deny them the right of association.

Only in situations where this would render assisted dying practically unavailable should such an institution be required to offer it.