You have had many posts of advice and so I would like to add my twopennorth. The thing is that this wedding , whilst important has raised many problems that you see in your life in general. The wedding itself I agree with others that it would be far less stressful for you to tell them that you are unable to go but you will be sitting thinking of them at the time and wishing them well, that you hope to have a small celebration with them when they come home, and you might tell a white lie and say that the reason is medical, but nothing for them to worry about. Well it wont be a lie as anxiety is definitely a major problem for you.
So I see that this situation has meant that you have faced quite a few things that you probably have known about but not addressed before. So for yourself, it really would be good if you could try and find help with your anxieties. If you can get some help and counselling you could work out what and why is the major problem and perhaps come to a better understanding and be able to deal with things a bit better and feel more in control. To do this without a specific event looming in the near future would be better and less stressful and you could move at your own pace.
But it has also thrown up some things that you would need to face. As your dogs are so important to you, it would be good if you could think about what would be best for them if there was a situation where you had to suddenly be away from home, such as an accident or had to be in hospital for a short time. Do you already have someone you trust who could feed them and take them for walks? Perhaps you have a friend who also has dogs or has had them in the past, who you could organise some emergency help with? If this is not possible, then perhaps you might need to think of looking into what local kennels you would be most satisfied with in an emergency. You could look around and take recommendations from other dog owners. Then it would be worth trying them in the kennels for perhaps two days, and see how that goes and then repeat that again after a couple of months. This way , you would get peace of mind, should there be an occasion in the future when you need to make a quick decision. If that goes well you could then go on to put them in the kennels for another couple of days and go away yourself, just for a couple of days, perhaps to the coast or to somewhere special to you.
These small steps taken at your own pace might mean that you feel less anxious and more empowered as you are the person who decides when and where. If you found yourself able to do this, in itself it may lessen your anxiety in that area and also you can look at it as a good example of what you have been able to do and then possibly employ the same small steps in other areas of your life. I do understand having had both anxiety and depression at times, and when I was in those situation the thought of doing what I do now , would have been unbelievable to me. I found the key was , in my own time, to see it as a choice I was making, not being forced into something by other people. Control is my key, with a certain amount of acceptance for what is possible and what is not worth the stress. I hope this is of some use to you, and the fact that your daughter does not feel as you do, means that she cannot understand what she is asking of you. Does she have any phobias that you know of? Perhaps not now regarding the wedding, but later you might ask her how she would feel if say a spider (if that was her phobia) walked across her pillow or down her wedding dress. She would no doubt be very shocked by that thought. Then you could explain to her that that is the way your anxiety affects y ou physically and mentally and that you constantly have to cope with this and it is not by choice. You may in the end let her see how much effort you have to put into things that she finds easy and she may have more understanding and sympathy. The best of luck