First off, what do you mean you installed Composer manually?
Also, which shared hosting provider are you using? Most of them support limited SSH commands, including Composer.
If for some reason you can't use Composer in the production environment, I recommend the following:
- Install PHP (of the same version as the production environment) to a local development machine (your desktop or laptop)
- Install Composer
- Do the steps needed to install Tinyboard on your local machine
- Copy your vendor directory to your production environment
- If you can proceed with installation on your production environment and you don't log any errors, it should be a success and you should be able to delete/remove the local development environment installation
As a side note, it's always good to run a near copy of your site locally so you can work on it safely and deploy it to your production environment without issues. (I do it for this site although I've let a couple bugs slip through on accident)