1) Firstly recheck that you have correctly configured .ics feeds URLs for exporting.
Open Booking > Settings > Sync > "Export - .ics" page and be sure that there you have specified URL for exporting .ics feeds with one level of depth
For example: https://server.com/ics/my_resource/
or if its not work try to set it as this URL
2) Please be sure that you are having .htaccess file in your website root folder.
3) Recheck about the any conflicts with some other plugin or actual theme.
Please try to deactivate one by one all your active plugins at the WordPress > Plugins menu and make testing after each plugin deactivation. If its not help, please deactivate all your plugins and active the default WordPress theme. And then retest it again.
4) Finally please recheck your error.log in your server configuration, about any relative errors. If you can not find error.log file, please contact support of your hosting company about helping in finding this file.
P.S. Please note, Booking Calendar generate virtual ics feed form the php script. So it’s not the real file at that folder. It’s generate such feed dynamically, when someone access such url.
System have to work in the same way as WordPress working with pages at your website. As you may know, you can have link to some page like this: https://server.com/category/some-page/
But its does not mean that you have real file like category/some-page/index.html at your server.
Your system access general index.php file at root of your website and then load specific page based on permalink structure.
The same with .ics feeds in Booking Calendar, when server try to access url: https://server.com/ics/principala/wpbm.ics
this .ics feed generate dynamically from the php file.