Gravitational binding energy of a part of system

Question

I want to calculate the gravitational binding energy of a small central cube (length $l$), which is part of a much larger cube (length L). I have the mass and gravitational potential distribution available for the entire system, just the small cube (internal system) and just the large cube-small cube (external system). Also, consider that all mass is gas, and it is possible to supply energy to the small central cube directly to make it gravitationally unbound.

Let's define some terms first: $\phi_{ext}$: potential due to just the external part (small cube subtracted from the large cube); $\phi_{tot}$: potential due to the entire system; $\phi_{int}$: potential due to just the small cube (in the absence of the external mass). I have attached a 2D representation of the system. The same subscripts also apply to mass $M$.

There are multiple approaches I can think of to calculating binding energy:

1. Consider the small cube (internal) and large cube-small cube (external) as two separate systems. First, calculate the energy required to separate those two systems until they are no longer gravitationally bound. Then add the energy required to break apart the small cube (no longer gravitationally bound to the external system). That would be:

   $E_{grav} = \sum_{-l/2}^{l/2} \phi_{ext}(x, y,z)*M_{int} (x, y, z)$+$\sum_{-l/2}^{l/2} \phi_{int}(x, y,z)*M_{int} (x, y, z)$

2. The second approach is to take the difference between the energy required to break the entire system and the energy required to break the external system. This should essentially be the small cube's potential energy in the entire system's presence. That would be:

   $E_{grav} = \sum_{-l/2}^{l/2} \phi_{tot}(x, y,z)*M_{int} (x, y, z)$

3. The third approach is to break apart the small cube in the presence of the external system. This would be computationally trickiest as the potential structure would change while removing every chunk of mass. Would this give the same answer as the first approach?

4. The fourth approach would be to offset the potential of entire system by the mean value of potential at the boundaries of the small cube, and then calculate the gravitational potential energy of the small cube. That would be:

   $E_{grav} = \sum_{-l/2}^{l/2} (\phi_{tot}(x, y, z)-\text{offset})*M_{int} (x, y, z)$

I would much appreciate it if someone could provide an insight into how to approach this problem. Thank you!