Local jws here including elders have had no problem going to the Catholic named hospital since the early 70's. No one calls them on it. For years....I supposed there are a few petty minded jws but they have no influence in this area. No more nuns working (or in hardly any Catholic hospitals, not enough new nuns, they are mostly old ladies in the old ladies Catholic home. Many jws HMOs are connected with this hospital so they would have to pay out of pocket to go to another. Most "mature" jws with money too find a way around this. It is a "conscience" matter. Since jws already have an issue with no taking transfusions and finding a doctor who will operate without one which are not "conscience" matters, around here they don't add another issue.
I find that "petty" jws abound in small, isolated congregations rather than cities with 10 or more congregations. The greater number of jws in the area tends to dilute the petty ones, especially since it has to go across at least 10 congregational boundaries. I have seen some congregation insist on white shirts, others not, others say double pierced ears is forbidden, others not. So you can just move across town and/or attend the congregation that supports your action.