I created a Laravel package (it is located in a private BitBucket repository) and I would like to test it in a new Laravel project if it works the way I want before registering it on Packagist. Is there any way to do that?
Yes, it is possible.
You have to add information where to look for your source code to composer.json. For example you could add your own private repository.
There are other possibilites, too. I suggest reading through the corresponding composer documentation