Data structure and algorithm for clipping triangulation with leaf nodes of an octree
Rather than making one spatial subdivision structure do double duty in representing both the voxels and the triangles, I would suggest creating a separate BVH for the mesh. You can find many articles and papers about BVH-building algorithms on the web. It's likely to be a more efficiently queryable representation of the mesh than the octree would be.

Given the BVH, it's easy to determine whether a given voxel might intersect the mesh by starting at the BVH root and traversing to child nodes that intersect the voxel box. Depending on the quality of the BVH, i.e. how tightly it fits the mesh, many voxels (or even higher-level octree nodes) may be able to be eliminated with only a few checks. For voxels that do intersect the mesh surface, the BVH traversal will reach down to the leaves, where you can accumulate the triangles into a list for clipping.

If the BVH is built in the mesh's local space, the mesh can easily be moved or transformed without needing to update the octree. (If translation is the only transform needed, then the BVH nodes will always be axis-aligned relative to the voxels, which simplifies the intersection test to an AABB overlap test; if more general transforms are needed, you'd have to use a more general OBB vs OBB test.)

However, note that a BVH doesn't lend itself that well to determining whether a things are inside versus outside a watertight mesh. If that's important to you (it wasn't clear to me whether it is), then you might want to look into using a BSP tree instead of a BVH. A BSP tree can be used for intersection testing similarly to a BVH, but additionally can represent the exact boundary of the mesh so that it can be used to determine whether a point (or voxel) is inside or outside. This property of BSP trees is widely used for collision detection.

An alternate approach to determining inside/outsideness is to use raycasting and count the number of intersections with the mesh (even = outside, odd = inside). The raycasting can be accelerated by the BVH as well.