How could a situation whereby the Roman Senate takes on the role that the English Parliament had, to varying degrees: a check on the monarch on behalf of the upper class and gentry?
I think one of the main problems with such a scenario is that during the Empire, there never was much of a separation of powers, due in large part to the Roman insistence that they weren't a monarchy. They'd more or abandoned the idea of separate powers for the executive and legislature, just dividing them (or not) as seen fit at the time.
So, ironically, it might have been the insistence that Rome was still a Republic that prohibited a vital feature of republics, even in nominal monarchies: a balance of powers.