sounds like the tech does not know what he is talking about.
of course they support downloading binary.
if you are going to use filezilla then you must set it to always download in binary. most other ftp programs do this by default. downloading in binary is the preferred way to download everything.
ascii only works for text based files like php /html/ etc. and will ruin true binary files like zips/jpg/ etc.
however, binary will not harm text based files at all.
so, use binary always and be safe.
now, since apparently your host does not know what it is doing, I would suggest you do find a different ftp program so that there is no concern about that.
then, use that ftp to download the complete phpbb files to your computer.
use phpmyadmin or whatever your host provides for exporting your database.
save that sql file to your computer as well.
once you have those two things, it won't matter what else you do , you will always be just a few clicks away from a complete recovery if things go south.
there is no need to create any kind of "mirror" for this.
even if the new board crashes and you lose it all you will still have the backups from the old one on your computer and can use that to start over.
however, remember to keep good backups of the new board as well. how often you do that depends on how busy your board is. if you only get a few posts a week then backing up weekly would probably work just fine.
if you get hundreds of posts per day then daily backups are what you would need to do.
oh, and there is no need for a new domain name for any of this process unless you just want one.