Here's a link to a picture of the caliper (unfortunately, it's super close up, so it might not help too much). I can get a better picture this evening if necessary.
https://goo.gl/photos/aYxQsVFkdi8hysZC8
So, as far as opening/closing the nipple as quickly as possible, what I do is, I get the preliminary bubbles out of it by opening it a little longer to get the bulk of the bubbles out. After I don't see any more bubbles, that's exactly what I do. Basically flip the bleed nipple open/closed a few times (with the tube still attached). I also make sure that the fluid is still moving OUT when I close the nipple, so air doesn't have an opportunity to get in as hydraulic pressure runs out.
Yeah, this is stumping me. Both my bikes I've been working on pretty steadily over the last month or two, but this is the last issue I can't seem to get over. I ordered a mityvac, which I've heard works well for pulling the old fluid out of the line, although, i'm not sure how much of a difference it will make. I've also considered doing the whole "reverse fill" method with a syringe, pushing the fluid in from the bleed nipple, but I wasn't sure what the opinion was on that, and wanted to see what people here said before I pulled the trigger on any other ideas.