Calling ilfilosofo / Austin Matzko,
I've experimented some more and looked at the code. Both when the next schedule time is displayed and when it's saved, the following expression is used to determine or set the time:
$time + (get_option('gmt_offset') * 3600))
The problem is that this always ADDS the GMT Offset and keeps advancing the time whenever I click "Change".
Also, when a names timezone is selected on the General options page (new feature), such as "Brisbane", the gmt_offset option is CLEARED (the time zone is saved in the option 'timezone_string'), which means the backup will run at the wrong time, unless manually adjusted.
A workaround I'm using now is setting the gmt_offset to 0 manually. It's incorrect, but I'm hoping it'll work.