Is this a Hamiltonian or a Lagrangian? Usually the Hamiltonian is equal to the total mechanical energy and I think that would require the two minus signs to be plus?

You could leave the spatial coordinates un-changed and set without changing the value of the Hamiltonian / Lagrangian. This results in conservation of energy.

The other common transformation in classical mechanics is:

After you examine the units of G I expect that you would find this leaves the Hamiltonian/Lagrangian un-changed and would result in something being conserved.

What would you expect to be conserved? I would expect:

Total mechanical Energy

Linear momentum of the centre of mass

Angular momentum of the system