Misiek so do you have any ideas how to fix it? :/
Fix what? I don't know that anything is broken.
What you are essentially asking is: I want to drive a Porsche 911, how can I make it use diesel fuel?
If your local gas station only sells diesel you may have a problem with your Porsche, but there is no way Porsche can 'fix' that. Even if Porsche had announced a 911 diesel at some time in the future there is no point in 'fixing' your problem now, at least not for Porsche. Wait for the 911 diesel or swap the engine yourself. If you want to choose the latter option, you should bring the experience to do so.
Coming back to Flarum and composer:
Flarum has been built around composer, and that for a good reason. You may circumvent that requirement during installation by e.g. installing Flarum locally and transfering the files per FTP to your shared hosting. But if you want to update your installation composer becomes essential again to sort out the various dependencies of the components Flarum is made of.
The makers of Flarum expressed their intention to develop a version of Flarum where this dependency management by composer is built into the software. This however does not have the highest priority during the beta stadium of this software.
PS: If you still want to swap the engine and don't lack the knowledge to do that, you may try to safe a dump of your database, reinstall the whole forum and then insert the contents of your database dump back into the new installation. As long as the database structure doesn't change, this might work. But I've never done that.
My advice is still the one I gave here: Pollux