Qualifying conscience protections for institutions with requirements that they minimize hardship caused to the patient would prevent religious institutions from acting as a choke point on the path to services.
Julian Savulescu's writing on conscientious objection is guided by an emphasis on the principle of distributive justice that does not allow religion to have a special status as justification.
To participate in a lethal injection is to occupy the medical role and use medical training for a purpose that is not part of the goals of medicine and that harms the recipient of treatment.