@canadanna, is the second photo the tree where you want to relocate the birdhouse? I think it's too close to the ground.
with plantladylin that the birdhouse should be 6 - 10 feet from the ground, and since you seem to have a problem cat, placing the nest about 10 feet high on a pole should keep the birds and your plants fairly safe from the cat.
I really wanted to say you should move the cat.