Tiling on slab or backer board
I hate to tell you that I know from calling the manufacturers of both hardiebacker and durock that they do no support concrete installations.  Both indicated that it would probably work fine but they simply haven't tested it.  

I have personally used it in a very small area that I needed to raise - very small.  You want to treat it like tile (thinset with large trowel) then hit each sheet with 4+ tapcon screws (which are hard to get just right).  This 2'by4' area took me way too long and I am not sure how I could have made it faster.  It has held up in a rental for 6-7 years now with no issues.  

I would think before doing this.  Also I would choose durock since it is a little more flexible than hardie (I prefer hardie for everything except floors).  

Given you are raising this area with the carpet, there is simply no logical transition.  Normally you would just buy a metal tile transition "L" and lay the flat side towards carpet... This will leave you too high.  The only thing that I can think of - and maybe others have ideas) is to basically raise the carpet area towards the tile, basically building a mini ramp.  Roofing shingles and plywood will get that job done.  Let's be clear, building a ramp so your carpet meets your tile OK is pretty ghetto.  It might turn out OK, but you will know.  

After all of these workarounds to get everything lined up, that shouldn't be I would have to ask the question - wouldn't you be better off demoing the tile in the bathroom and laying the tile on the slab?  Demoing tile isn't that hard and laying tile on a slab is the simplest install.  Depending on size of bathroom this may be the least time consuming thing.  I know you have a bit more in material cost but this is a permanent solution and the right way.

The crew I flip houses with have a simple motto - we do hard stuff, we do some stuff a bit ghetto, but we never make the ghetto stuff hard.