Rocket, the osteopath was good for us too, well it is hard to say, but it seems that things started getting better from then onwards. Could be just coincidence though. Are you going to go there again? We went twice overall. First time she said he was all scrunched up on one side, she said she'd tried to release the tension. Second time was just to check, she said things seemed good, and we needn't come back. Bless her honest soul :)
Regarding pumps, I have an electric avent pump, however switching to a double pump was a relevation. We picked one up from the local surestart centre, we rented it, for £5 for 6 months. Yes, that's a total of £5. Was not Medela but rather Ameda I think, but I think they're actually the same thing (perhaps Medela aimed at end-users, Ameda aimed at institutions or something). They're the ones they tend to use in hospitals. Admittedly, it's not very dignified to be sat there doubly attached, but well, I didn't care about dignity anymore, just about the sleep time it gained me. Also, I got more milk that way.
That said, many people find it more efficient to bf on one side whilst expressing on the other at the same time. It helps with the let-down, and is nice to have it both done at once. But I never managed the trick. It involves rugby-ball hold for the baby, which just never worked for me.
Hmm suckling laziness... you say she is not suckling at all? Is she creating the vacuum at all? We had times when he wouldn't go on properly (i.e. no vacuum) and then things are just a bit pointless. Perhaps you could try hand-expressing a little, getting the let-down going, and then sort of enticing her to go on... Don't know.
One thing our lactation consultant suggested to deal with him always falling asleep within (what felt like) seconds after latching on, was speed-boob-swapping. She said that with the initial let-down, the milk comes strong and fast, but then it evens out, and lazy baby might be too lazy (or too tired) to suckle to keep it going. So, as soon as baby is not doing right anymore (suckle suckle SWALLOW suckle suckle SWALLOW ifyswim) you take her off, put her on other side, and start again. Repeat. In time, she will get used to doing it right. (I found it hard to actually take him off. Finally he was on! Why take him off now? But I had to admit that he was not really on anymore, and it was actually pointless to just keep going... so, took him off and put him on other side although I knew it would be a struggle)
Also, she suggested to undress him. Suckling at the breast is so comforting, so add to that nice warm coziness, so of course he falls asleep. So for a while he would be just in his nappies to feed. It did help to keep him alert, I think.
One thought I might add: You say how you are trying to get her to latch on whenever she is awake. I can understand where you are coming from. But, and this is purely from my own experience: I found that if DS was not at all hungry, he wouldn't latch on, or wouldn't suckle. Just wouldn't. I found it very frustrating. HV comes along, asks me to feed him - wrong time, he won't. And it looks like I really haven't got a grip on the latch... when in fact the problem lies elsewhere.
Anyway, when he was not feeding despite screaming with hunger, that was one thing. But if I understand correctly, your daughter is now getting enough food. She is not starving. She may be sleepy and not feeding as often as you expect, but this does not now mean she is going hungry. When we were in that situation, it was important for us to feed when we were both relaxed and calm, AND he was starting to be hungry. One day I decided to de-stress. Not do anything else but concentrate on DS only. I watched him closely and offered him the breast whenever he made the slightest pretence of rooting, but calmly, not insisting - if he didn't want, he didn't have to. It took a while, maybe 5 or even 6 hours since the last big feed (which had actually been by tube) but then he did go on, and suckle, and feed. Not enough, I still had to top him up, but he did feed. That did wonders for my confidence and my nerves!
Do you offer her breast before feeding her by other means? If you do, I don't think you need to try to get her to bf in between. Just try to get the balance right, not to soon (she won't be hungry) and not too late (she'll be too upset to figure out something new). That means watching her carefully. Do you know the signs to look out for? (I'm not implying in any way that you're not doing that already, btw. Just asking.) It also means maybe letting go of some schedule you have been advised to use (like, every 3h or every 4h), at least temporarily. You know now that she's not starving. It doesn't matter if she has a few 'late' feeds as long as she doesn't go mad hungry.
One last thing: There are all kind of guidelines out there of how much, how often, how long a baby should feed. However, the ONLY way of judging if a baby is getting enough, is weight gain and general health. So. If she is gaining weight, she is getting enough. Knowing that might help you relax a bit (and relaxing is sooo conducive to better BF...). Don't let anyone pressurise you into feeding her more, or more often, based on guidelines... if she is gaining weight she is getting enough.