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I think the don't have an answer is no. If I'm not mistakento your full question, but I can say something about the existence of such permutation representation.

Assuming a decomposition into frames existed, it cannot be preserved by the action of Co0, because it would yield a subgroup of Co0 of index at most 4095 as a stabilizer of a frame, or equivalently, a subgroup of Co1 of index at most 4095 as a stabilizer of a distinguished 24-tuple of the 98280 pairs of antipodal norm 4 points in the Leech lattice. However, according to the ATLAS, the largest maximal subgroup of Co1 is the group Co2, of index 98280.
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I think the answer is no. If I'm not mistaken, the existence of such a decomposition would yield a subgroup of Co0 of index at most 4095 as a stabilizer of a frame, or equivalently, a subgroup of Co1 of index at most 4095 as a stabilizer of a distinguished 24-tuple of the 98280 pairs of antipodal norm 4 points in the Leech lattice. However, according to the ATLAS, the largest maximal subgroup of Co1 is the group Co2, of index 98280.