*File > Export Selection* might be useful for what you want.

There's no need to hide any layers, just make sure everything you want to export is grouped. If you want to export a background as well, you will need to use a filled rectangle, and also group that with each object.  Using *Export Selection* ensures that the export is based on the selection only, even if the groups overlap each other.

Here's an example:

Each asset here is a group (including a background rectangle). I selected each group in turn, and did *File > Export Selection*.

[![enter image description here][1]][1]

You can also use the Asset Export panel and click and drag assets into it. Then you can export directly from there.

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/3GH4d.png
  [2]: https://i.sstatic.net/kt17s.png