If the monorail isn't running, then there should be a bus. Buses start running at 6:30AM, but they can be very infrequent that early. I'd check to see if the monorail was running at 7AM, and, if not, I'd head straight to the bus stop.
I don't have experience from BLT, only Poly and GF, but the experience should be the same. For an Epcot PPO ADR, I've used taxis and targeted to be in line before the tapstiles by around 7:30ish. I am a big fan of taxis in this very situation - quick, effecient, no wait. You can judge your travel time much more easily than with a bus at that hour.
I don't have experience from BLT, only Poly and GF, but the experience should be the same. For an Epcot PPO ADR, I've used taxis and targeted to be in line before the tapstiles by around 7:30ish. I am a big fan of taxis in this very situation - quick, effecient, no wait. You can judge your travel time much more easily than with a bus at that hour.
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