Without loss of generality, consider minimizing
$F(x,y,z)$
subject to constraints
$G(x,y,z)=0$
$H(x,y,z)=0$
Multiple constraints can be reduced to a single one:
$G^2+H^2=0$
Then, Lagrange method generates equations:
$ {\partial F \over \partial x} + 2 \lambda (G {\partial G \over \partial x}+H {\partial H \over \partial x}) = 0$
$ {\partial F \over \partial y} + 2 \lambda (G {\partial G \over \partial y}+H {\partial H \over \partial y}) = 0$
$ {\partial F \over \partial z} + 2 \lambda (G {\partial G \over \partial z}+H {\partial H \over \partial z}) = 0$
When comparing this to equations produced with canonical method with multiple Lagrange multipliers, it seems that each of the component is proportional to this scalar $\lambda$. That is
$\lambda_G = 2 G \lambda$
$\lambda_H = 2 H \lambda$
I wonder if this observation is mathematical folklore.
The problem is, that your transformation of the constraints will render constraint qualifications unsatisfiable. In particular LICQ is never satisfied, since the derivative of $G^2 + H^2$ is zero if $G = H = 0$.
Moreover, if $G = H = 0$ is satisfied, you obtain $\lambda_G = \lambda_H = 0$ with your approach.