Transfer operator spectrum of the discretized harmonic oscillator

I’m reading "An introduction to quantum fields on a lattice" by Jan Smit.

In chapter 2, the transfer operator ?̂ is defined and shown to be equal to
$$
hat{T} = e^{-omega^2 hat{q}^2/4} e ^{-hat{p}^2/2}e^{-omega^2 hat{q}^2/4}
$$
(There is another question about this section: Eigenvalue spectrum of the transfer operator for the harmonic oscillator)

With the usual coordinate representation:
$$hat{q} to q , ~~~ hat{p} to -i partial / partial q$$

The coordinate representation of the ground state is given by:
$$
langle q|0rangle = e^{- frac{1}{2} sinh tilde{omega} , q²}
$$
$ tilde{omega}$ and $omega$ are related via
$$ cosh tilde{omega} = 1 + frac{1}{2} omega^2$$

Now the question:
I don’t know how to derive this equation:
$$
hat{T} |0rangle = e^{-frac{1}{2} tilde{omega}}|0rangle
$$

What I thought of so far:

• To bring $hat{T}$ to the form $hat{T}=e^{-hat{H}}$ by using the Baker-Campbell-Hausdorff formula.
  But the exact BCH, $$ e^{X}e^{Y}=e^{{X+Y+[X,Y]/2}} $$ is not applicable here, since $[X,[X,Y]] neq 0$, for $X = hat{p}^2$ and $Y = hat{q}^2$.

• Use the coordinate representation of $hat{T} |0rangle $, e.g.
  $$ e^{-omega^2 hat{q}^2/4} e ^{-hat{p}^2/2}e^{-omega^2 hat{q}^2/4} e^{- frac{1}{2} sinh tilde{omega} , q²}, $$
  where the last part is the ground state.
  Then I use $ e^{-hat{p}^2/2} = 1 – frac{p²}{2} + frac{p^4}{8} + …$ and then insert the coordinate representation of the momentum operator, giving a ladder of derivatives of zeroth, then second, then fourth order… But this also does not yield the desired result.

I am thankful for any ideas!