Woud it be possible to test the soft minibb on localhost with Mamp
I never tried that, but I'm mostly sure it could be possible, if it's about Apache/mySQL/PHP.
To carry on the installation I have to register a registration code on www.minibb.com
Not sure what do you mean. You don't have to register any codes on miniBB website to test the installation or even run the live forum. It's yours for free with no registrations or extra deals.
Did I should redo the whole installation ?
You could configure and test it on localhost even with the pre-filled database.
Then, you only should transfer the database to your new hosting, transfer all files, changes all necessary permissions, and configure miniBB to run with the new database and the domain. You don't have to re-install it, just transfer it properly.