notice that this is not the same as being a toric vector bundle. also, i'm not interested in the projectivization. rather, i'm perfectly fine working with smooth quasi-projective toric varieties.
examples: are the total spaces of $O(2)\to P^1$, or $O(-3)+O(1)\to P^1$ toric varieties?
i want to argue that then second is not, at least not as a symplectic quotient (i understand the algebraic geometry definition is more general): since $H_2$ is one-dimensional, it must be realised as the quotient of $C^4$ by $U(1)$, imposing one momentum map; i cannot find any that would produce the desired result. in particular, using weights $Q=(1,1,1,-3)$ gives the total space of $O(-3) \to P^2$, which contains it as the normal bundle to one of its $P^1$'s.
i was pointed to the paper https://arxiv.org/abs/math/9911192 but there lemma 1.1 is about the projectrivization of a bundle.