I think, that in order to answer this question it is worth to conisder the complex algebraic analog of this question. Namely, suppose we have a $\mathbb C^*$ action on a projective manifold $V^n$. Can we find an invariant Lefshetz pencil?

It is sufficient to conisder the case of (complex) surfaces to spot some problems. Namely, the action of $\mathbb C^*$ in a neighborhood of an isolated fixed point should be of one of the following $3$ types:
$$(z,w)\to (tz,tw), \;\; (z,w)\to (tz,t^{-1}w),\;\; (z,w)\to (tz,w)$$

where $t\in \mathbb C^*$ and $z,w$ are local coordinates at the fixed point.

There are not so many examples of $\mathbb C^*$ actions on surfaces that have only these three types of fixed points. But here are two examples: first is $\mathbb CP^2$ with the action that fixes one point and a separate line. Second is the action on $\mathbb CP^1\times \mathbb CP^1$ that fixes $4$ points. In both cases there is an (obvious) invariant Lefshetz pencil. You can also take the first example and blow up several distinct points on the fixed line in $\mathbb CP^2$. Maybe this is the complete list...
Surelly, from all these examples one gets also the symplectic Lefshetz pencil of the kind that you wish to have.

I don't see why there will be much more examples if you will go into symplectic cathegory.