Laravel: Is it possible to log stack trace when exception caught, and continue the execution?

You can simply call your app's exception handler manually:

<?php
use App\Exceptions\Handler;

try {
    foo();
} catch(\Exception $e) {
    $h = new Handler(app());
    $h->report($e);
}
bar();

This gives you the same error reporting you'd normally get, including any customizations you may have made for additional logging channels, etc.

Laravel has a helper method just for this, see the rescue method.

return rescue(function () { 
    return $this->method(); 
});

Using your example it would look like:

rescue(function () { 
    return foo(); 
});

bar();