I could have written your original question a couple of months ago!
I'm pretty large up top, too, in the neighborhood of a 40H, give or take. I had all these fuzzy romantic dreams of sidelying in bed, sleeping through those first few tough weeks, and just being glowing super breastfeeding mom. Ummm...no.
We had some struggles at first (some really bad ones for a while, actually) and the first two weeks were full of ups and downs. Nursing was a team effort with my husband helping, getting nursing pillows just right, sometimes propping up her head with more pillow/towel, and all sorts of various helping aides. I tried lying down and there was just no way I could do that early on. People would say "oh, just put a little burp cloth under your breast to support it." No. I don't need a burp cloth. I need *scaffolding*! ;)
By two weeks we were through a good portion of the learning curve. By six weeks, we were through the classic learning curve and doing great. I felt confident, I NIP'd all the time, I still used pillows for support but no longer needed too many extra helpers. I did still need a c-hold because these 40H's have a gravity of their own and my DD couldn't hold the breast in her mouth well enough against that kind of force. ;)
I would try periodically, but it still wasn't quite right. Sidelying involved a lot of shifting, maneuvering, setting it up just so, and so forth. That isn't what I wanted in the middle of the night. So, I kept up with sitting up in bed. I would wake a good bit, but in time and with practice could almost doze sitting up in bed with her. I keep a pillow vertical on my headboard. Another pillow to sleep on that becomes a support for her when she wakes/nurses. Book light on her cosleeper to give me just a little bit of lighting to help her latch.
A couple weeks ago when she was 3 months old, we started trying again and she's really getting it now! She has essentially full head control, can support herself pretty well on her side, and all around just is bigger (actually, she's huge! :) and can handle more stuff on her own. She latches pretty much by herself now. I still mostly use a c-hold to help her latch onto the planetoid that is my breast, but occasionally, in some positions, I can let go. I figure in another month or three, she'll really be able to hold the breast herself and my arms/hands will finally be completely free.
So, we've been practicing and she's been loving it as much as I. Sometimes she lays flat on the bed and turns towards my breast. Not the best way for swallowing, but she's old enough now that I'm not as concerned with it as she seems to be intrigued and delighted by the position. Other times, we lay tummy to tummy. Sometimes she can support herself there just fine, sometimes I pull my husband's pillow up behind her so she doesn't have to focus on keeping herself on her side.
I can nurse either boob (top or bottom) in those positions. It varies which works best. Haven't figured out the subtle differences in why it works better/worse from time to time, but I'm sure we will.
She positively loves the position and gets all excited/flappy when she sees we're about to do that. It's adorable! We actually both fell asleep nursing the other day in that position and it was heavenly! She frequently falls asleep like that now, and once she's done I often fall asleep, too.
It still takes some support on my part, holding my breast in a modified c-hold. This can be a pain since it means my upper arm is often busy/having to work against gravity, but I'm getting better at it and have found variations that make it easier. In another couple weeks I bet we'll be able to reduce that. This work, though, does tend to keep me awake a bit. And, just watching her/being fascinated with this new "trick" also delights me so much it's hard to sleep until she's done. :)
She's now about a week away from 4 months...so it might take you a while as well. But keep trying and I bet you'll see baby step improvements along the way. We're only doing this during the day or for naps right now. Not yet ready to try to do this in the middle of the night, sleepy, maneuvering in the dark. We will be soon enough though and I know it will be wooooonderful. Good luck and know that it probably can be done, but is another serious learning curve for many of us.