Let's say I have a symplectic manifold $\Omega$ and two smooth time-dependent hamiltonian functions $H$ and $\tilde{H}$ defined on it. I would like to study the time-dependent canonical transformations $$ \Phi_{t,0}^{\tilde{X}}\circ(\Phi_{t,0}^X)^{-1} \quad \text{(†)} $$ where $X=\lbrace\cdot,H\rbrace$ and $\tilde X=\lbrace\cdot,\tilde H\rbrace$ are the time-dependent hamiltonian vector field of $H$ and $\tilde H$, and $\Phi_{t,0}^X$ is the hamiltonian flow of $X$ from time $0$ to time $t$.
More precisely, I am studying constrained hamiltonian systems and H and $\tilde{H}$ are two equivalent (total) hamiltonians, their difference is null everywhere on the constrained surface, thus on the image of the solutions. Because of this it is possible to show that on the constrained surface $$\lbrace H_t,\tilde H_s \rbrace=0 \quad\text{thus}\quad [X_t,\tilde X_s]=0 \quad \forall t,s .$$
In the time independent case I think the expression $$ \Phi_t^{\tilde{X}}\circ(\Phi_t^X)^{-1} = \Phi_t^{\tilde{X}}\circ\Phi_t^{-X} = \exp \left(t \tilde{X}\right) \exp \left(-tX\right) = \exp \left[t \left(\tilde{X} - X\right)\right] = \exp \left[t \, \lbrace \cdot,{\tilde{H}-H\rbrace}\right] $$ should be correct (vector fileds are here identified with elements of the Lie algebra of $\textrm{Diff}(\Omega)$, and the result can be seen as a particular case of the Baker–Campbell–Hausdorff formula).
What I'm looking for is an analogue of this for time-dependent hamiltonians.
To this purpose I've tried to use the results [3.1, 3.3, 3.6] in the paper "A Lie Group Structure on the Space of Time-dependent Vector Fields" by A. Posilicano. I planned to express the flows in (†) via exponentials of elements in the lie algebra that Poilicano introduces - and maybe try to see if these commute with respect to Posilicano's Lie brackets $[\![\cdot,\cdot]\!]$- then use BCH formula. The problem is I cannot figure out the expression of the Lie algebra elements necessary for writing (†).