Sorry to play “pile on” here, but I am experiencing a similar issue. When exploring away mode, I saw your plugin reported my local time as being 0103 when in fact it is 0803.
I checked my WP general settings and the time zone is correctly set to Los Angeles and my local time is accurately displayed in general settings. As an experiment, I changed my timezone to a random selection and that change was reflected on the away mode page.
I experimented with changing the time format, standard AM/PM to 24hr time and apparently it makes no difference to BWPS.