quantum field theory - Non-relativistic QFT Lagrangian for fermions

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter and D. Walecka in Quantum Theory of Many-particle Systems, page 19):

$$\hat{H}~=~\int\hat\psi^\dagger(\mathbf{x})T(\mathbf{x})\hat\psi(\mathbf{x})d^3x$$ $$ + \frac{1}{2}\iint\hat\psi^\dagger(\mathbf{x})\hat\psi^\dagger(\mathbf{x'})V(\mathbf{x},\mathbf{x'})\hat\psi(\mathbf{x'})\hat\psi(\mathbf{x})d^3xd^3x' \tag{2.4}$$

The field $\psi(\mathbf{x})$ and $\Pi(\mathbf{x})=i\psi^\dagger(\mathbf{x})$ ($\hbar=1$) satisfy the usual canonical quantization relations, but if I try to build a Lagrangian as:

$$L=\int\Pi(\mathbf{x})d_t\psi(\mathbf{x})d\mathbf{x}-H.$$

It turns out that, because:

$$d_t\psi(\mathbf{x})=-iT(\mathbf{x})\psi(\mathbf{x}) - i\int\psi^\dagger(\mathbf{x})V(\mathbf{x},\mathbf{x'})\psi(\mathbf{x'})\psi(\mathbf{x})d\mathbf{x'}.$$

If I combine both expressions the Lagrangian turns out to be zero (a proof of the last equation can be found in Greiner's Field Quantization, chapter 3, it can be derived using $[a,bc]=[a,b]_\mp c\pm b[a,c]_\mp$).

My questions are:

1. What is wrong in this derivation?

(Greiner manages to get the Hamiltonian from the Lagrangian but he makes some integration by parts that he gives as obvious but that for me should have an extra term)

2. How can you derive $$\frac{\delta H}{\delta\psi}=-d_t\Pi$$ from the previous hamiltonian? From this expression, the Euler-Lagrange equations can be derived easily, but I can't seem to find the way to get it.