I use a Diva, which is the same shape and size, I believe, as the Keeper, so hopefully what works on mine will work for you.
I usually pinch mine just a little to break suction and don't try to fold it at all. I tilt it, keeping the opening up, so that one edge comes out before the other. This kind of pushes the edge into an oval that slides out pretty easily.
The other thing I'd recommend is to play around with different insertion folds and see if you can not only do them on the way in but also on the way out. My Diva came with instructions to fold the cupin sort of a C shape for insertion. I learned from other cup users on the internet two more folds. The one I use is the 7 fold. Flatten the cup like you do for the c-fold, but instead of folding it edges together the second time, you take one corner down so that the edges don't line up. The edges form a 7 when you look at them from the front (not the top view, which is where you see the C of the C fold). I wish I could post the pictures here, but the one site where I know there are pics isn't responding right now. The other one I know is the punch down. Put one finger on the edge of the cup and push down so that that bit of the edge tucks inside the cup. This is I think the least intrusive of entries, but it also takes a little longer to coax it open once it's inside, and I don't really need it quite so small, so I use the 7 as the best compromise for me.
Now, if you use any of these on the way out (I'd think you could do the punch down, if you just reach up fairly high to get at the edge then hook your finger), keep in mind that you probably won't want to let your cup get too full, as the change in shape might press fluids out of the cup.
If you can't figure it out, bump the thread again in a day or two and it'll remind me to get the link to the pics for you. I can't imagine the site will be down much longer than a day at most.