Yeah, working with the bezier mask can be a bit tricky. Key is to make sure you're using the right selection mode. Start by clicking somewhere in the Pan/Crop workspace other than on the mask so none of the mask is selected. Then add your keyframe. Then hover over the part of the mask you want to move -- if you want to move just an anchor, hover over a point; if you want to move a segment (the line between two anchors) hover over a segment; for selecting the whole path, you can hover over any part of the mask. Then right-click > Select > [whatever you're trying to select]. Then you can just click and move whatever you want.

To switch to a different mode, follow those steps again.

Also note that if you've selected a segment, to move that segment you have to grab one of the anchors on it. If you grab the segment, you'll be editing its curve.

Hope I interpreted your question correctly.

Rob