> - check whether JSBSim is actually applying the weight for the fuel

Yes, we are. Look at FGMassBalance.cpp:

bool FGMassBalance::Run(void)
{
  if (!FGModel::Run()) {

    Weight = EmptyWeight + Propulsion->GetTanksWeight() +
GetPointMassWeight();

    Mass = Weight / Inertial->gravity();

// Calculate new CG here.

    vXYZcg = (Propulsion->GetTanksCG() + EmptyWeight*vbaseXYZcg
                                       + GetPointMassCG()       ) / Weight;

// Calculate new moments of inertia here

    Ixx = baseIxx + Propulsion->GetTanksIxx(vXYZcg) + GetPMIxx();
    Iyy = baseIyy + Propulsion->GetTanksIyy(vXYZcg) + GetPMIyy();
    Izz = baseIzz + Propulsion->GetTanksIzz(vXYZcg) + GetPMIzz();
    Ixy = baseIxy + Propulsion->GetTanksIxy(vXYZcg) + GetPMIxy();
    Ixz = baseIxz + Propulsion->GetTanksIxz(vXYZcg) + GetPMIxz();


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