Yes, a Junior transfer can still be considered for recruitment. I know we've accepted them before since they can still be valuable assets to our chapter.
UCSC doesn't have Greek housing so your dog can just chill at your house with no problems.
Be sure to check out the OPERS Fair in the fall to meet representatives from all of the Greek orgs as well as many other campus clubs. That's when we do the most recruitment, so you'll surely see a ton of girls representing Panhellenic (Kappa Kappa Gamma and Gamma Phi Beta). This event is a few days before fall classes start.
I'm the Membership Chair for Kappa, so feel free to PM me if you have any more questions.