FlashFXP supports timezones but they can be somewhat buggy in v3.6, since you are a registered user if you're interested in trying beta software I would recommend you contact us to join the beta team.
You can read more about the v4 beta here
https://oss.azurewebsites.net/download.php?s=4
Also v3.6 has an option to preserve the date/time (in the preferences) on download but if the timezones are buggy for you then your results might not be very reliable.