Thank you for your question today, I look forward to assisting you. I have nearly 20 years of legal experience in various disciplines, including JAG.
Ok. Here is the problem. Under the SCRA, legally you would likely have to pay through the end of July anyway.
The SCRA rules allowing you to break a lease based on orders of separation states that the earliest you can be released from the lease is 30 days following the next regularly scheduled rent payment following your notice.
You gave notice on June 17, and I'm assuming that your next rent payment would have been due at the beginning of the next month...July. So, your lease break wouldn't legally be effective until July 30th under the SCRA and you'd have to pay through that period for rent.
That is important, because it basically takes the Department of Justice off the table as someone that you can use here.
If your lease says something different than the SCRA, essentially allowing you to break the lease more quickly, that is a matter of contract law and not of military law. That will depend entirely on the wording of your military clause within the contract/lease itself and your recourse would be to sue the landlord in state court (small claims most likely in order to reduce costs).
My reading of the contract is agreement with how you read it, but again, this isn't a military law issue. You'll have to threaten suit in state court to resolve the contract dispute, which is what this really is.