Are there any merge tools that offer features for this a lot more sophisticated than basic text comparison?

Not merge, but jq can diff, and it can also do a consistent dump (jq -cS) for you.

Edit: to clarify, jq -S does deep keys sorting.

  $ echo '{"z":{"b": "second", "a": "first"}, "x": 4, "y": 7}' | jq -S
  {
    "x": 4,
    "y": 7,
    "z": {
      "a": "first",
      "b": "second"
    }
  }

Powershell has Compare-Object, which will diff .NET objects. It has the convenient alias "diff". JSON can be converted to .NET objects by ConvertFrom-JSON.

So you can import 2 JSON files and diff them in Powershell.