Is there an easy way to sharpen a wire loop carving tool?
This answer is not based on first-hand knowledge of sharpening wire loops, but is based on extrapolating what works for paint scrapers.  

Many years ago while performing one of those early-marriage projects that required the removal of paint from second-hand furniture, I decided that my scraper blade needed to be replaced.  Ran over to the local hardware store and the knowledgeable clerk informed me that I did not need a new blade - just sharpen it - doh, really? how?

He proceeded to demonstrate and after thirty seconds and two swipes of a file to the cutting edge I went home.  I still have the scraper and blade, though it's much reduced in size.

I would take the same approach with these flat wire loops and use a file to restore the edge.  A few firm swipes should produce a nice edge.  It might also be possible to treat the edges the same way you would sharpen a pocket knife with a whetstone or with a small hand-held stone. It might be a bit tricky on the curved parts, but I think any artisan could handle a file to create a curved edge.