Moodiness of dementia patients. Any advice?

This is common. Stay for as long as it takes for you to feel like you have made an attempt at a visit. At this point they are like little children with a short attention span. My father will ask the same question over 3 - 4 times in a visit. When they swing so quickly from happy to angry, you cannot take this to heart. It is part of the disease.
Take comfort in the fact that you placed her in a place that takes good care of her and has such good staff.

If a longer visit makes her agitated cut the visit short. That you go at all is enough under the circumstances.

I wonder if she is lacking the energy of staying focused for more than a few minutes, so it makes her agitated. I would just follow her lead. I wonder if you were doing things around the room or sitting quietly watching TV with her if the visits could be longer. I know that a few minutes is not very rewarding for you and you would like to stay longer sometimes.

Have you tried going out and coming back in? Does she greet you with excitement the second time? If so, perhaps there is another person you could say hello to for a few minutes. Perhaps you could return a phone call or some other short task. Read a few pages in a book and then go back in. Saying hello and goodbye is easier than carrying on a conversation. I'm glad she knows you. You might try grooming her. Brush her hair or buff her nails. Or massage her hands. She might be willing to sit with you for the length of time it takes for the task. Play a simple card game. I'm sorry if these suggestions aren't appropriate for her present condition. You will feel better fir having seen her even if you can't get her to engage for long.

It sounds like Mom is glad to see you. It confirms she has not been abandoned. Good. Got that checked off the worry list for the day. Goodbye. Perhaps just knowing that you are still in her life is enough, and she doesn't have the energy/skills to make conversation.