AFAIK, the BA lounge in E closes for a couple of hours in the morning. The Oneworld lounge finder confirms, closes at 7am, reopens at 9am:
https://www.oneworld.com/airport-lounges?location=BOS
Personally, I don't think it's worth it either way, even if you did get there at 6 - keep in mind, Terminal E doesn't have pre-check in the morning when you'll be there. Logan has some decent food options. Even in the morning, I'd prefer to go sit down at Legal Seafoods over clearing security twice.