My special needs chickens live in one of my waterfowl coops with 7 ducks and 6 geese with no issues for the most part. However, their is one important piece of information you didn't tell us: How many geese and are their both males and females?
If you have both sexes, they will pair off and each will claim a nest area and aggressively defend it during mating season. I have to put dividers between my ducks/chickens and geese during breeding season to keep the gander from attacking any poor critter that wanders too close.
As to the dog kennel area....is it on concrete and tall enough to put some sort of top on it (wire, netting, metal roof)? If you have a secure top and bottom, and the fencing is chain-link or heavy wire, then a large dog house, or even one of those plastic dog igloos would work for a shelter. Either that or put in two walls coming out of the corner a couple feet each direction with a roof on it.
As long as the wire area around the geese is secure, their actual housing only needs to be a place to get out of the weather and make a dry nest. If you are worried about raccoons reaching through into the run, then zip tie some wire or hardware cloth (small holes) around the entire bottom up 3 feet anywhere their are no walls or other obstacles blocking their reach.
Good luck!