Hi Jennifer:
Walt Disney World buses do not provide Resort-to-Resort direct transportation, so some effort will need to be made to get to where you need to go. To be frank, the last time my family had a reservation at Cape May Café (breakfast) at the Beach Club Resort while we were staying at a "non-Boardwalk area" Resort, we just took a taxi to make it easier for us. I would budget 20 minutes taking a taxi from the Contemporary.
However, there are ways to get to the Beach Club Resort relying on Walt Disney World complimentary transportation,
which starts running one hour prior to theme park opening. You could take the monorail or walkway (about 10 minute walk) to Magic Kingdom, wait for a bus (could be a 20 minute wait, give or take) which will take you to Yacht and Beach Club Resorts (about 20 minutes) where you can find Cape May Café.
For me, for the extra sleep and convenience, I would consider taking a taxi. Some should be waiting outside of your Resort (or your Resort's concierge can arrange for one).
Enjoy your vacation.