The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated.
The Higgs field gives us interactions between chiral fermions and the Higgs, $yh\cdot \chi_\alpha\eta^\alpha$ which produces mass terms $m \chi_\alpha\eta^\alpha$ if the Higgs field has a vacuum expectation value $h=v$. We have $m=yv$.
However, in more general quantum field theories, one may write down similar (bilinear) mass terms manually without any Higgs field (and without cubic Yukawa terms) and none of these issues affects the fact that the spin-statistics relationship holds. The field $h$ is spin-zero field and bosonic, the fields $\chi,\eta,\Psi$ (the latter is the 4-component Dirac spinor combining the previous two) are spin-1/2 and fermionic fields.