I think it's more likely than not, in this case, that if you're on that 1st bus and explain the situation to the driver, he/she will either drop you at BC or arrange to have a bus take you from DHS to BC. There are actually lots of buses leaving DHS in the a.m. heading to the Epcot resorts to pick up Guests and take them to MK or AK.
Also, I don't know whether this has changed lately, but in the past if you called and explained that you were running late due to transportation challenges, the resort would give you a little extra time.