A form is used to logout:
<form method="POST" action="{{ route('logout') }}">
@csrf
<x-dropdown-link href="{{ route('logout') }}"
onclick="event.preventDefault();
this.closest('form').submit();">
{{ __('Logout') }}
</x-dropdown-link>
</form>
Unless bootstrap isn't loading correctly and interfering with:
this.closest('form').submit();
You can rewrite logic something like:
<div class="dropdown-menu dropdown-menu-right" aria-labelledby="navbarDropdown">
<a class="dropdown-item" href="{{ route('logout') }}"
onclick="event.preventDefault();
document.getElementById('logout-form').submit();">
{{ __('Logout') }}
</a>
<form id="logout-form" action="{{ route('logout') }}" method="POST" style="display: none;">
@csrf
</form>
</div>
Notice the change to document.getElementById.
But really a little trial and error here and you should be able to work it out.