If you drive your rental car to the Beach Club Resort for your ADR, you will not be allowed to leave it there while you visit the park, unless you valet park there, then you could walk or boat to either Epcot or DHS. I believe Valet parking is still $14 plus tips for the day.
If you choose not to valet park you will have to move your car from the Beach Club resort, you can then drive to any park you wish.
To get to the Beach Club resort without using your rental car for an early ADR I would recommend a taxi. Fare could be anywhere from $10 to $25 assuming you are staying at a Disney resort. Getting there via Disney Transportation could be difficult for a 7:00am ADR, and could take a considerable amount of time. If your ADR is say 8:00am you could use the bus by going to DHS and then walking or boat to BC, allow a minimum of 1 hour to get there.