Excellent question. Not a psychologist here (or in the US, I’m in Australia), but I practised as a clinical gerontologist for several years.
Firstly, some general thoughts.
In respect to areas where there are moral clashes, it’s helpful to keep in mind: (1) your professional licensing board’s regulations; (2) where you work and the organisational policies that govern your practice; (3) the relevant legislation in your jurisdiction (I assume in the US it’s primarily state and federal).
Depending on what action you take, you may fall afoul of either (1), (2) and/or (3), which could expose you to legal action (e.g. negligence) either by the client and/or your employeer in addition to whatever sanctions your licensing board may have.
If you feel uncomfortable with the case, you may refer the client to someone else and/or ask your manager to reassign the case if possible.
Secondly, in respect to LGBTQ policies. In general, I find that this crops up far less than is assumed: they have health professionals (whether counsellors, doctors, social workers, etc.) that specialise in their community and so their first recourse is often to them rather than to a general practitioner in the community.
Aside from that, there are no easy answers that results in both you and the client being mutually satisfied. I don’t see myself as setting my particular religious, moral or ethical values aside when I worked with a client, given that my values suffuse and enrich my practice, and I hope that this is for the benefit of my clients.
However, I did take care to prioritise the patient’s values and goals in respect to their health. It’s a “client-centered practice”, as it were. It’s not so much about what I think is best (or worst) and making a judgement on their decision, but about presenting the clinical information as accurately and intelligibly as possible to the client to ensure that they’re able to weigh up the evidence, their own life circumstsances and so arrive at an informed decision. Sometimes this involves a patient making a decision with which I would vehemently disagree, but nonetheless I respected.
A common example I came across in my work was regarding the discharge destination of elderly patients who had unsuccessfully undergone physiotherapy/occupational therapy after an accident. It was clear, sometimes, that it simply wasn’t safe for them to go home due to their limited mobility and the lack of care arrangements. The only viable place for them to go was into a nursing home. Many of the patients disagreed, many of their children disagreed, and much of the time the patient said “I’d rather die at home tomorrow by falling over than live out the rest of my days in a death factory”.
I disagreed vociferously, but after negotiation and discussion it was something that I accepted: their life, their health, their decision.