that's quite a bit of code!
however I can see from library/widgets.php that the theme or theme creator puts in a lot of custom code for widgets, in a way that overlaps with how Widget Logic works. (A very immodest guess would be it builds on WL's code, though doing much more impressive things than I've ever been bothered with of course!).
so it'll take me a while to work out just where the clash is - it's much larger than my modest plugin, sorry.
i have another suggestion, as a possible workaround - try 'Enable accessibility mode' under screen options on the appearance/widgets admin page. It might be just the 'ajax' saving of code that breaks, and the old fashioned "submit form" still works