I knwo this may not be the answer you want to hear, but you may just have to go with it and keep encouraging her to take more. Hopefully she will catch on. I don't think there is a problem as long as you aren't putting her down if she falls asleep on the breast. If she does fall asleep, gently wake her and then do wind down with her. It will likely save troubles in the future with sleep troubles.
Also, it sounds like she's doing pretty well, what happens if you don't offer after her A time? Will she make it through to her next feed? Maybe you need to drop down to a 2.5 hr EASY or something for a bit until you can stretch her to the full 3 hours.