Exception Handler on a Switch
There is nothing wrong with rethrowing an exception of a different type, the problem lies with how you are doing it. According to the PHP docs, you can pass a previous exception as a second argument to the new exception. This preserves the original stack trace. I would also provide a new error message as well:

throw new MyException("A new error summary", $e);


A new take on your switch statement:

public function render($request, Exception $e)
{

    switch (true) {

        case $e instanceof Exceptions\Auctions\AuctionTypeException:
            throw new Exceptions\Notifications\AlertException("An auction type failed to save", $e);
            break;

        case $e instanceof Exceptions\Auctions\AuctionArgumentException:
            throw new Exceptions\Notifications\AlertException("An auction argument was not valid", $e);
            break;

        ////////////////////////////////////////////
        // Custom errors to display in error view //
        ////////////////////////////////////////////
        case $e instanceof Exceptions\Notifications\AlertException:

            return response()-&gt;view('errors.notification', ['message' =&gt; $e-&gt;getMessage(), 500]);
            break;

        //////////////////////////////////////////////
        // Unknown exceptions are rendered normally //
        //////////////////////////////////////////////
        default:
            return parent::render($request, $e);
    }

}