If it's a choice between miles or sleep, do your morning miles maybe once or twice a week and get your sleep the rest of the time. Regular daily bike commuting at a pace you're used to and have been doing for awhile shouldn't interfere with recovering, and I don't think you need to stop doing it for the recovery days. You can make a point of doing it slowly if you feel you need to. I think it's good to be used to getting on the bike every day. But beating yourself up at the expense of sleep is not a good long term plan. It's one thing to do it sometimes, or to do events that involve sleep deprivation, but going without a good night's sleep on a daily basis is not healthy, not good for training, not good for productivity, mood, concentration, memory, etc, etc, etc.
The other thing is, your trip is a month away. There is only so much more ready you're going to get in a month. I'm not saying quit training and sit on the couch, but if you're feeling generally beaten down, now is the time to rest up and recover.