One thing I don't follow is your 1/4" height calculation. I don't
see how you're going to get the hardiboard, thinset, tile, etc in and
only raise the floor 1/4"

If you can do the tile work yourself and get the tile done or about
the same as vinyl, I'd go with the tile.

I meant that the floor would be a max of 1/4" higher than the
surrounding floors. Currently in the hallway adjoining the room, we
have 5/8" wood flooring installed. Adding 1/4" hardiboard, thinset,
and tile would probably raise it to approximately level with the
adjoining rooms.

I think I have reconciled myself to the vinyl sheet floor - it's not
as expensive as I had previously supposed ($1.50 / sq ft is the
highest I found for the flooring itself), so it's a little easier to
consider. I just plan to use the type of flooring that can be attached
around the perimeter using fiberglass tape - roughing the surface for
better adhesion isn't an option, which I have heard is required if you
want to use the other type of floor.