Opajaap I was having this same problem and seeing as this isn't resolved I'll make a comment in hopes you can crunch it. Also I'm using Firefox on Linux so might have to take that into consideration too.
Anyhow I think the problem lies in the jQuery click even to show/hide the table. I couldn't find the code but I'm going to assume by the way it was acting there was a
preventDefault() in there somewhere. That might solve the problem or it might not (block level elements within other block level elements aren't always the best practice I've found out). You could also fix it similiar to how I did. In the page wppa-settings.php down by line 420 you have the entire table heading wrapped within an <H3> tag where I think the root problem lies (in some browsers perhaps anyhow). I got your plugin to work by doing changing this (code truncated):
<h3><?php _e(); echo(' '); e_(); wppa_toggle_table(); ?>
to something like this:
<h3 style="display:inline-block;"><?php _e(); ?></h3> <div style="display:inline-block;"><?php e_(); wppa_toggle_table(); ?>
You could also wrap the entire code block in it's own div but you still have to make the elements inline-block (this should prevent them from breaking up too bad on smaller screens).
I can now save all my settings and the layout doesn't change that much (a little more vertical spacing between the tables but doesn't look bad).
Hopefully this will help with the issue. By the way - impressive plugin found it incredibly helpful thank you!