@alexsteele let me see if I can help you regarding where to place every piece of custom code.
earlier in my process I created a new trait, called hasBrandPhoto - it was stored in jetstream/src/...
For this, you can create a Traits folder under app folder for example. Then you can create your trait in there (make sure you change the Namespace accordingly).
created new file: vendor/laravel/jetstream/Http/Livewire/UpdateBrandInformationForm.php
This should go under app\Http\Livewire\ folder (which you may need to create if you don't have it yet). Again, update Namespace to match the new location.
Import that new form into here: vendor/laravel/jetstream/stubs/inertia/resources/js/Pages/Profile
This shouldn't be needed in your case as you chose to work with Livewire instead of Inertia.
Registered the component here: vendor/laravel/jetstream/src/ JetstreamServiceProvider/php
Do this at the boot method in the JetstreamServiceProvider located under app\Providers instead of the one at vendor folder.
Beware although that you are registering 2 different clases for the same component, so the second one it's the only one it will work (if I'm not wrong).
Also, check that you are importing the App\Http\Livewire\UpdateBrandInformationForm instead of the vendor one. Actually, delete it from the vendor folder as you shouldn't have created that file there in the first place.
I think I've covered all your custom files but I might be missing something else.
I've made most of this steps in a fresh local project and got it working without touching the vendor folder so I believe it should work for you too.
Hope this helps!