From my experience there is no BART near OAK, you have to take the BART Bus to get to the BART. And this BART station is not a place where you want to spend anymore time than you have to
Please let us know if you have prepaid for this hotel, as I agree that you need to consider other options - if possible. The forum members are here to help you - please listen to their advice.
I'm visiting SF in October and have found the hotel prices to be higher than I expected, so I am searching for a vacation rental. There are many available and the prices can be a lot less than a hotel. For 3 people you can find a nice vacation rental for $100-$150/night.
Riding the BART is a good experience - but I wouldn't want to have to do it every day of my trip.