I'm going to agree with DrJean and say that you shouldn't take the dog. 7 weeks is a LONG time to live with a muzzle and/or leash attached to the pup at all times.
What kind of dog is it? If it's been trained to chase cats, it's probably not going to be untrained any time soon. Maybe with some intensive training the dog could learn to control its prey drive (Because that's what your brother is encouraging by asking her to chase small animals), but int he meantime, there's going to be huge amounts of stress for everyone involved.
Before we rescued her, my dog hunted small animals in the neighborhood. Now after 6 years she lives peacefully with my five cats (and formerly guinea pigs and a rabbit), and she can watch a squirrel or a rabbit in the backyard without chasing it. She is 100% alert, and completely intrigued - I have no doubt that she'd take off in an instant if I said she could, but as it is, we no longer even have to coax her "Sasha... nooo..." anymore. And she's a chow/sheperd mix, both of which have fairly high prey drives. So it's possible, but it took a lot of work. Though she was never taught to kill anything, it was a necessity.
Especially if this is a bigger dog, I'd tell them that you're just not comfortable having a dog who is a potential danger to your kitties. Maybe they can find someone with a dog, or a petsitter as suggested, but if not, a kennel really is the best option. I know they may seem cold, but Jessie will get human interaction every day, and she won't have to wear a muzzle. She'll be as well as a dog can be when away from their family for an extended period of time, and you won't have to worry about anything happening to your cat/s.
And afterall, you could always visit Jessie if you want to give her some TLC.