Just a FYI...there's not really a need to separate the bottom two sections. I realize that if it was me of course I'd like to get them apart but it won't affect the performance at all and you can wash out that air chamber under running water and shake out the water and reassemble and it's all good.
You could try using something like those rubber pads that you use to open jars so you can get a good hold on it.
I haven't taken my bottom section apart since I assembled it on day one.. I'm sure I will at some point.