I would go the MailStore Home-route. With that software you can get the messages you have been collecting in your FossaMail client . While it may not say that it supports FossaMail directlyy, Fossamail is a fork of Thunderbird and that one is supported in MailStore Home.
When that import is done, MailStore Home can export your messages in different formats. You can choose one that is supported by The Bat (.eml) if there isn't an option to import your content into Tjhe Bat directly. Mail filter rules and your contact book are not backed up in MailStore Home, but exporting those in a .csv file from FossaMail is often the easiest way to get contacts imported into another mail client.
MailStore Home is freeware. Not sure if you could use 'Mozilla Backup' (freeware last time I checked) in combination with FossaMail, but you could give that a try too.