Lots of options you can consider. If you want new it would most likely be a hybrid at that price. Used you might have more options, but my experience in the bay area is that everyone wants way too much money for a used bike, especially when you consider what you can find new right now.
Sounds like you want a bike that has attachments for a rack as that may be needed for a child seat.
Touring bike would be nice and stable, but it also might be long, a possible negative for BART.
Don't worry to much about handlebar, you can swap out for a narrower one if needed. If you go with a flat bar you might even be able to cut it down, though I would suggest findng a bike you like and seeing if the handlebar length is an issue before you do anything about it.
You might consider seeing if any of the REI Novaro brand bike work, as they ahve a 20% off coupon right now.
If commute into San Francisco, consider Public Bikes. They have a big sale on the 2013 bikes right now (in store only) as they are moving from two locations to 1 bigger one. You can get a few models for under $500 right now. Might work well for a short commute like that.