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Full Idea
One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums.
Gist of Idea
A nominalist might avoid abstract objects by just appealing to mereological sums
Source
E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4)
A Reaction
I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity.