Some of the information on here doesn't match my experience. Macmillan or Marie Curie can be great, but it can take a while to get appointments for financial and other information, also if you rely on them for care, what they can offer is limited by who else needs them that week, so you might end up with one six hour time slot. Marie Curie nurses come and sit with the terminally ill person and are very nice, but not really experienced counselling types, my experience anyway.
Similarly, hospices are amazing places, but the one I have spent the most time in was busy with nurses going about their business and although they are very kind and check you are ok, they are not in the business of counselling you. The hospice phoned me every few weeks afterwards, the lady was very nice, but she was a volunteer.
Really high quality death support and counselling is not something you just get in the system at all, not in my experience anyway. People are lovely, kind, sympathetic, the odd counseling appointment may be available, but they are very busy, I think having a more sustained interaction and support, and joined up guidance could be very helpful from someone who has been there and knows the systems. Also, the dying person themselves might like to have another familiar face around, hospice nurses and carers work on shifts and don't sit (in the hospice I know) for long periods if ever just chatting with the person. The chaplain calls, does a prayer, moves on.
I think it's a great idea OP, I am not entirely sure what your duties would be, and it could be a bit fraught between the family and medical staff if it weren't a straightforward case, but otherwise, I do see a role for much more family and support of the person.