I had the exact same problem when my daughter was born this past May. I fed her exclusively on one side for awhile while the other nipple healed. I also pumped on the side with the damaged nipple--just with a little hand pump, even though I had an electric Madela one--because it was easier to just do that. Could you try putting Lanolin all over that nipple before you pump it? That's what I did. And I can honestly tell you that although I knew I was supposed to pump every time I used the other breast, I didn't. I pumped twice, maybe three times a day on that side, which I felt really guilty about, but I was just so overwhelmed with everything it was hard to find the time to pump on that side when my daughter wanted to be held all of the time, and I wasn't coordinated enough to do it while she was eating. So perhaps you could try pumping less frequently on that sore side until it gets better? I know that may not be advice a LLL leader or breastfeeding consultant would give you, because I know you need to keep up your milk production in that breast, but it worked for me. I brestfeed exclusively on one side for several days, and just pumped a couple of times a day on the other side, and let that side MOSTLY heal before offering it to her again. It took way too long for it to completely heal to wait that long. When I offered her the damaged nipple, I just gradually did it, only one or two feedings a day at first, and sometimes only for the first few minutes of a feeding before I would switch her, and then I gradually worked back up to using it as much as the other breast. This worked for me--I was lopsided for a little while, but it wasn't too bad, and that other breast seemed to catch up pretty quickly. Once it was completely healed, I used it more often than the other to try to catch it up, and everything seemed to work out just fine. Another thing I would suggest is using different breastfeeding positions--that's one way to help with nipple soreness. And I found one position worked better on the side with my damaged nipple than the other positions.
Good luck--it WILL get better, and it will heal--just stick with it! Mine is fine now, and my daughter has no problem breastfeeding off of either breast--I think it gets easier when they get a little bigger.