Yes, your layer should be an Embden, I went thru the same thing wondering if I had a Pilgrim pair but my 7 yr olds are two gals, an Embden and a Toulouse.
Put your lil poopers (dog) on a leash or crate train her, then you don't worry about watching her and only allow her outside within a pen she can't get out of.  It works if you make an attempt and you don't have to watch them all the time, just make a simple routine of where and when they are allowed freedom.
I live w/four six hunting dogs, an ACDx and a BCx, a wolfx and two very old golden retrievers between my adult kids and I, not counting the eldest two son's having absolutely wonderful pitbulls, one each...between all of us coming and going, we've only lost one bird to my 3rd son's dog and that was his fault, not his dog's.  I was pretty mad at my 25 yr old son, it was one of my prettiest banty hens, a partridge rock.  The dog ran around the chicken pen, jumping and barking and the hen flew out.  
I'm sorry about your goose, it made me laugh that you figured out what sex it was.  I'm trying to figure out what my nearly one year old 4 Am Buff geese are, hoping for at least one gander.
I'd use a clean sponge or washrag in warm water that baking soda is added to for gently soaking and keeping the wound clean.  Its what I used all these years on sled dogs that had wolf in their line.  Those type of dog always wanted to fight and they'd bite the forelegs of the other dog, trying to break the leg so there would be puncture wounds that had to heal from the inside out in order to not fester.  It does not sting, either since our bodies are mostly salt water.  Betadine and peroxide are very harsh but have their uses and are needed when its fungus or festering.  Otherwise, the baking soda water is kindest and works well.  Big hugs to your family and your remaining goose as it will be without its mate now.