I don't think the distinction in their governance is important for a map. There's a slippery slope somewhere where osm becomes wikipedia. What's important is that there's a hardware store, and the name it is known by.
I would use franchise. If you want to make up "weak_franchise" for stores that are part of Ace but for which this is not really apparent to the public, that seems ok. Around me, there are several that are "[townname] Ace Hardware", and you know you are going to Ace, but that it's locally owned. There was "Robinson's Ace", where you knew it was Ace, but now it's "Robinson's True Value" and it's not really different. I don't like the co-op tag because normally co-op has the connotation that the user of the store is a co-op member, not that the store is at a higher level.
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