A quick and dirty way which works.
If $B,D,F$ are known, the problem reduces to a linear regression. So, make a three dimension grid and for each triplet compute the sum of squares until you find a minimum. For the best triplet, recompute $A,C,D$ and start the nonlinear regression.
Because of symmetry, you must not compute all points. Suppose that you start the search at $B=V$ and you want to perform $N$ evaluations using a step size of $DV$, then the structure of the loops would be something like
DO IB = 1 , N-2
B = V + (IB - 1) * DV
DO ID = IB+1 , N-1
D = V + ID * DV
DO IF = ID + 1 , N
F = V + IF * DV
compute SSQ by linear regression