Regularization for inverse filter design
This alternative approach seems fishy to me. Consider the two-dimensional case with $C$ having eigenvalues $\lambda_0, \lambda_1$ that are real and positive and $\lambda_0 \ll \lambda_1$. You have
$$
\det C = \lambda_0\cdot \lambda_1
$$
What you want to prevent with the regularisation is that your inverse gets a too big eigenvalue $\lambda_0^{-1}$, i.e. you want to cap that to at most $\beta^{-1}$. Meanwhile, $\lambda_1^{-1}$ is fine. But if you put in the corresponding eigenvector $v_1$ into that alternative formula, you get
$$\begin{align}
  Hv_1 =&amp; \frac{\det C}{|\det C|^2 + \beta}\cdot\operatorname{adj}C\: v_1
   \\  =&amp; \frac{\lambda_0^2\cdot\lambda_1^2}{\lambda_0^2\cdot\lambda_1^2 + \beta}\cdot C^{-1}\: v_1
   \\  =&amp; \frac{\lambda_0^2\cdot\lambda_1}{\lambda_0^2\cdot\lambda_1^2 + \beta}\cdot v_1
   \\  \approx&amp;
          \frac{\lambda_0^2\cdot\lambda_1}{\beta}\cdot v_1
\end{align}$$
I.e. the scaling for all inputs depends on $\beta$, not just for the low-eigenvalue ones.
I suppose this formula might still work in some situations (perhaps when the determinant is mostly imaginary?), but it certainly isn't appropriate for the general case.
Use Tikhonov.