Relation between Jordan-Wigner transformation and Hilbert-Schmidt inner product

It's a good question, but the answer is that the Hilbert-Schmidt inner product and the Jordan-Wigner transformation are not the same, even for the special case of fermionic Hamiltonians.

First let us consider spin-5/2 particles (they are still fermions). The fermionic Hamiltonian in this case will be $6^n times 6^n$ matrix for $n$ particles, so for one spin-5/2 particle we have a 6x6 matrix which cannot be decomposed as a Kronecker product of 2x2 Pauli matrices.

Now let me also mention that the Jordan-Wigner transformation is not the only way to convert a fermionic Hamiltonian into into a "Paulinomial" (polynomial of Pauli matrices). There's also, for example, the Bravyi-Kitaev transformation which will give a completely different Paulinomial compared to the Jordan-Wigner transformation, even when starting from the same fermionic Hamiltonian in operator form. Yet another example separate from the Jordan-Wigner and Bravyi-Kitaev bases is the parity basis (see Eqs 20-21 here).