OK, so is it possible the hand-wringing anxiety is about making a wrong decision and looking foolish?
This is an absurdly compressed version of the sales training I used to deliver. We don't have three months of full-time contact to work on it 
Attention
Interest
Desire
Conviction
Action
Attention:
This is for establishing a communicative atmosphere. Narcissists generally appreciate a compliment on their appearance and a little question about it - 'You look great! Is that a new haircut?' type of thing. As your person ('P') arranged the meeting, perhaps approve of his venue choice.
Say you're really pleased you ('we') are going to find a way to sort things out. Get him to agree ("You must be, too" or similar.)
Interest:
Engage P in the process. Say it's a difficult situation for both of you/everyone, and you respect his views or his input is obviously vital.. Ask him to explain what he's thinking. Listen very carefully. Feed back his points to show you've understood. Establish common ground: there will be some, and you should clearly indicate which bits you agree with. Don't disagree with anything at his stage, just listen properly. Make it very obvious you understand his particular concerns, and that you respect his viewpoint.
Desire:
The purpose of your meeting is to get the problem resolved. Begin by expressing the solution in a way that fits his perspective. If, say, you want Mum to go into a specialist facility and he wants her to stay at home, the 'solution' might be: We need to figure out how best to keep Mum safe and comfortable, bearing in mind her health and cost considerations. Ideally, he should be enthusiastically agreeing to this (or at least have stopped wringing his hands!) Itemise the considerations (not proposed solutions), getting firm agreement at every point.
Conviction:
This is the longest part. Go through your 'desire' points, discussing how the various available solutions fulfil them. Aim to build a pros and cons list for each of the points. Don't be afraid to admit to cons from your solutions - each of these will be a "we need to find an answer to that", not an "argh!"
It's possible there is a third way, or an add-on to your preferred solution, which will successfully resolve each of the cons.
Refer to P's wisdom, compassion, knowledge, whatever her relishes in himself, while seeking his input at every step. Get agreement all the way through. If he's agreeing to your statement "You've got a point, the care home can't provide X, can it?", that's better than a negative response. You want him to be constantly agreeing with you, while you move the argument by stages towards your solution.
Use your evidence to support your arguments, and also give attention to his. While presenting your evidence, make sure you keep showing how it answers the concerns raised in 'desire'. Ask P, "You can agree that this provides X, yes?" and "Do you feel this is close to what you wanted?" - getting agreement as often as possible.
If he goes into long explanations, shut up. Listen carefully and interject with understanding or agreement wherever you can.
Action:
You must come away with a defined action, even if it's not a signed contract or whatever you're aiming for. He might need to seek professional advice: specify what it is, and exactly when you'll meet again to discuss it in detail. He might be all emotionally discombobulated, in which case your action will be to soothe him, arrange an exact time to meet again, and offer to supply any more information he wants.
Love-bomb him when you part. If he's stressed, buy him a coffee & cake. Tell him what a fantastically constructive meeting it's been, how much you appreciate his thoughts, and so on. Don't forget to pay him a personal compliment again. Say you'll ring him in the evening/tomorrow/Monday, and do it.
... Is this type of thing likely to work with your P?