The facts have already been stated that refute your what if. Whether you missed it or not, you're arguing for a perpetual motion machine.

The Layman's version of the 3 Laws of Thermodynamics are:

    1) You can't win
    2) You can't cheat
    3) You can't even choose not to play

For instance, take a Brayton thermodynamic cycle. It's motor cycle, but you can literally just reverse the arrows on this cycle, it becomes a reversed Brayton cycle, which corresponds to a heat pump (or a refrigeration machine, depending on which inlet of the machine you're interested in). In practice, the reversing of the arrow means the reversing of the fluid flow, which means compression processes are now expansion processes and vice versa.

Carnot theory does still apply. The COP (Coefficient Of Performance) of a Carnot heat pump is 1/n, where n is the usual Carnot efficiency of a heat engine.

With n=1/3, you get a COP of 1/1/3 = 3 = 300%

Carnot efficiencies consider ideal machines. So ideal Carnot batteries have a n * 1/n = 100% efficiencies. It means you can't increase the efficiency of a heat engine by putting a heat pump in front of it. In an ideal case (ignoring all losses and inefficiencies), you would get the same efficiency. In practice, you can only be worse.