The answer to your question is yes, given that the Hamiltonian perturbation indeed is sufficiently small. Conceptually the idea of the construction is better formulated as follows: fix the symplectic form $\Omega$ and instead deform the fibration by a smooth (in general non-symplectic) isotopy in order to obtain a different monodromy. (If you instead insist on fixing the fibration, the symplectic form will be deformed by the pull-back under this smooth isotopy.)

First, construct a standard symplectic neighbourhood of $F=\pi^{-1}(1) \subset (E,\Omega)$ being the product $$(F \times [-\epsilon,\epsilon]^2,\Omega|_F \oplus dx \wedge dy).$$ We can moreover use an identification in which all $F \times \{(x,0)\}$ are fibres $\pi^{-1}(z)$ for $z \in \partial D$ in the base. However, all fibres for $z \notin \partial D$ can only be assumed to be $C^\infty$-close to the symplectic surfaces $F \times \mathrm{pt}$ (equality for all fibres would imply flatness of the fibration).

Second, take a smooth isotopy which deforms $(u,(x,y))$ to $(u,(x,H_x(u)+y)$ , $u \in F$, for a smooth function $H_t \colon F \to [-\epsilon,\epsilon]$. It is readily checked that the monodromy around $\partial D$ is deformed by the corresponding Hamiltonian flow. Note that you have to interpolate this smooth isotopy to make it defined on all of $E$, and also do this in away so that the image of a fibre is still symplectic. The latter can be done, but I am a bit brief at this point.

The answer to your question is yes, given that the Hamiltonian perturbation indeed is sufficiently small. Conceptually the idea of the construction is better formulated as follows: fix the symplectic form $\Omega$ and instead deform the fibration by a smooth (in general non-symplectic) isotopy in order to obtain a different monodromy. (If you instead insist on fixing the fibration, the symplectic form will be deformed by the pull-back under this smooth isotopy.)

First, construct a standard symplectic neighbourhood of $F=\pi^{-1}(1) \subset (E,\Omega)$ being the product $$(F \times [-\epsilon,\epsilon]^2,\Omega|_F \oplus dx \wedge dy).$$ We can moreover use an identification in which all $F \times \{(x,0)\}$ are fibres $\pi^{-1}(z)$ for $z \in \partial D$ in the base. However, all fibres for $z \notin \partial D$ can only be assumed to be $C^\infty$-close to the symplectic surfaces $F \times \mathrm{pt}$ (equality for all fibres would imply flatness of the fibration).

Second, take a smooth isotopy which deforms $(u,(x,y))$ to $(u,(x,H_x(\phi^t_{H_t}(u))+y))$ , $u \in F$, for a smooth function $H_t \colon F \to [-\epsilon,\epsilon]$. It is readily checked that the monodromy around $\partial D$ is deformed by the corresponding Hamiltonian diffeomorphim $\phi^\epsilon_{H_t}$. Note that you have to interpolate this smooth isotopy to make it defined on all of $E$, and also do this in away so that the image of a fibre is still symplectic. The latter can be done, but I am a bit brief at this point. Also, you need to be careful concerning the support of $H_t$.