I also doubt they would let you in with a boarding pass to another concourse.
Two options:
Get an Admiral's club day pass.
Cost= $50
Buy a refundable ticket on an airline flying out of the J terminal so that you can get airside, and then cancel it after getting to the lounge. Delta is in the H/J complex and you should be able to get buy a fully refundable one way MIA-ATL ticket for sub-$400.
Cost = $0, aside from the inconvenience waiting for refund to clear to your credit card, and the hassle of walking from D to H/J and back.
Safe Travels.