Such modules do exist for $p>3$. The Lie algebra $L_p={\rm Der}\ H(2;\underline{1};\Phi(\tau))^{(1)}$ has a filtration of depth $1$ such that ${\rm gr}(L)$ is a graded Lie algebra of type $\rm H$. This can be seen by embedding it into $W(2;\underline{1})$ as explained in Strade's paper in Can. J. Math. , vol. 43, 1991, 580-616. As $TR(L)=2$ the Lie algebra $L_p$ contains a $2$-dimensional torus, $T$ say. Any $2$-dimensional torus of $L_p$ must intersect trivially with $(L_p)_{(0)}$ because any nonzero torus of $(L_p)_{(0)}$ is $1$-dimensional and has no zero weight on $L_p/(L_p)_{(0)}$. L_p/(L_p)_{(0)}$ (here $(L_p)_{(0)}$ is the zero component of the filtration). Passing to corresponding graded algebras and modules and using some results on representations of $H(2;\underline{1})^{(2)}$ mentioned in Jim's answer one can see that all but two of the irreducible restricted representations of $L_p$ are induced from irreducible restricted representations of $(L_p)_{(0)}$. The equality $L_p=T\oplus (L_p)_{(0)}$ then shows that all possible $T$-weights (including zero) occur in such modules with the same multiplicity.
As a historical remark I should add that irreducible representations of finite dimensional graded Cartan type Lie algebras of type ${\rm W, S,H}$ were first studied in the late 70s by Ya. Krylyuk, a former PhD student of Kostrikin. Unfortunately, the results of his PhD thesis were buried at in the cemetery of "secondary scientific and technical information" called VINITI and remained completely unknown to the rest of the world (the was no ${\rm arXiv}$ at the time). Wikipedia now has a comprehensive article about VINITI.
