Jk as a pre-mixing method
The 3D-Var cost function including the Jk term can be written:
\[J(x) = J_b + J_o + J_k = \frac{1}{2} (x - x_b)^{\rm T} B^{-1}(x - x_b) + \frac{1}{2} (y - Hx)^{\rm T}R^{-1}(y - Hx) + \frac{1}{2} (x - x_{LS})^{\rm T} V^{-1}(x - x_{LS})\]
Setting the gradient to zero, we have at the optimal $x$:
\[\nabla J = B^{-1}(x - x_b) - H^{\rm T}R^{-1}(y - Hx) + V^{-1}(x - x_{LS}) = 0 \]
or
\[\left[B^{-1} + V^{-1} + H^{\rm T}R^{-1}H\right] \left(x - x_b \right) = H^{\rm T}R^{-1}(y - Hx_b) + V^{-1}(x_{LS} - x_b). \]
Equivalent pre-mixed first guess
Assume now that $\widetilde{x_b}$ is some yet unknown, pre-mixed field depending on $x_b$ and $x_{LS}$ that we want to determine. By adding and subtracting identical terms to the gradient equation, we have
\[B^{-1}(x - x_b + \widetilde{x_b} - \widetilde{x_b}) - H^{\rm T}R^{-1}(y - Hx + H\widetilde{x_b} - H\widetilde{x_b}) + V^{-1}(x - x_{LS} + \widetilde{x_b} - \widetilde{x_b}) = 0,\]
which, when reorganized gives
\[\left[B^{-1} + V^{-1} + H^{\rm T}R^{-1}H \right] \left(x - \widetilde{x_b}\right) = H^{\rm T}R^{-1}(y - H\widetilde{x_b}) + B^{-1}(x_b - \widetilde{x_b}) + V^{-1}(x_{LS} - \widetilde{x_b}). \]
If the last two terms on the right hand side add up to zero, i.e.,
\[B^{-1}(x_b - \widetilde{x_b}) + V^{-1}(x_{LS} - \widetilde{x_b}) = 0, \]
which means that
\[\widetilde{x_b} = [B^{-1} + V^{-1}]^{-1} ( B^{-1} x_b + V^{-1} x_{LS} ), \]
then we see that by using this mixed first guess the Jk term can be omitted, provided we use a modified B-matrix with the property that
\[\widetilde{B}^{-1} = B^{-1} + V^{-1}. \]
By writing
\[B^{-1} + V^{-1} = B^{-1}(B + V)V^{-1} = V^{-1}(B + V)B^{-1} \]
we easily see by simply inverting that
\[\widetilde{B} = [B^{-1} + V^{-1}]^{-1} = B(B + V)^{-1}V = V(B + V)^{-1}B. \]
To conclude, a 3D-Var minimization with Jk is equivalent to a minimization without the Jk term, provided that one pre-mixes the two first guess fields according to
\[\widetilde{x_b} = [B^{-1} + V^{-1}]^{-1} ( B^{-1} x_b + V^{-1} x_{LS} ) = \widetilde{B}( B^{-1} x_b + V^{-1} x_{LS} ) = V(B + V)^{-1}x_b + B(B + V)^{-1}x_{LS} \]
and use the following covariance matrix for this mixed first guess:
\[\widetilde{B} = [B^{-1} + V^{-1}]^{-1} = B(B + V)^{-1}V = V(B + V)^{-1}B. \]
Whether this is implementable in practice is a different story, it just shows the theoretical equivalence, and how LSMIXBC should ideally be done if Jk is the right answer.