I need to use bsdtar to create an archive and set the modification time the same way as I would do with tar (with --mtime=date option), but I cannot find any equivalent option for bsdtar. I need to use bsdtar because with tar I have issues with extended attributes. Any suggestions?
If you don't absolutely have to do this as an argument passed to bsdtar, you can change any file's mtime with touch (requires root privileges): touch -d "<time>" <filename>.
See https://askubuntu.com/questions/62492/how-can-i-change-the-date-modified-created-of-a-file
Otherwise this seems to be impossible to do in the exact way you described.