Instead she will throw a tantrum and berate my Father until he agrees she does not have to go. This behavior by her and my Fathers response helps neither of them. When I try to defuse the situation I become my Mother's target. How can I get my Mother to the Dr. so she can receive the care she needs?
When you tell her focus on the lunch and not the appointment, "Mom, the appointment is at 11:30 a.m. and then I thought we'd go to that little place that has those huge hot fudge sundaes."
The day of the appointment make sure your mom has plenty of time to get ready. Don't rush her. Try to stay away from saying things like, "We have to get going if we're going to make it on time." If she putters around her room take a couple of blouses from her closet and ask which one she'd like. Then leave her to get dressed. Each time you check on her do something helpful to keep her mood light. Bring her a small glass of orange juice (and to check on her progress) and encourage her to have some juice.
Do everything you can without crowding her, stressing her out, or making her feel rushed. Make her morning pleasant and enjoyable and stress-free.
Can you get your dad on your side? You mentioned your father's response. I don't know what his response is but it sounds like whatever his response it may not be helping the situation any? Let him know what he needs to do (or not do) for your mom to be successful in getting to the Dr. Does he need to be away from the house that morning? Does he need to stand back and let your mom get out of the house on her own (but supervised) time?
When your mom is agitated, as you mentioned, let her burn herself out. Don't jump into the fray with your dad as it will only agitate your mom more. Get your dad out of the line of fire and remove yourself from the situation for a few minutes. Give your mom time to calm down. After a few minutes go back to your mom and see if you can't get her back on track.
Whatever obstacles there are in place keeping your mom from the Dr. (your dad, his reactions, time, etc.) remove them as best you can.
Do you know why your mom doesn't want to go to the Dr.? Is she afraid? Does she not want to go out? Understanding why your mom reacts so strongly to going to the Dr. may help in any future attempts to get her there.
Once you've gotten her to the Dr. come back and let us know how you did it as this is a common problem.
If your mom is narcissistic and getting dementia, it may be a very challenging battle ahead, especially, if you dad is not helping matters. What's wrong with him? Why won't he support you? Having to fight them both to get mom care sounds like a lot of work. Are you the Durable POA or is it dad? I think I might read a lot about how things often go in these situations and consider if I want to take it on. I bet plenty of adult children with similar situations will chime in here. It can be very time consuming and painful.
I had to insist on medical treatment with my cousin who was a rather spoiled and stubborn person, but with patience and some planning, I was able to get her help. And as her dementia progressed, she grew kinder and easier to work with. I don't think that's common though.