Gday,
I would lay a plywood sheet floor over the tile before laying the underlay . That way the tiles are not going to show though as the carpet wares. Glue fix and screw the sheets into the concrete substrate.
Depending on the height of the skirtings. Removing them prior to installing the ply floor may be advisable. Re install them on top of the new ply. That way, when the carpet is installed the skirtings will match the rest of the house. At the doorways it is easy to create a hidden ramp to match the different heights between the floors I'm a carpenter/ builder by trade and have done this many times over the years both in domestic and commercial applications. It basically comes down to what quality job you are looking for.
With the cost of the carpet. How long is a piece of string? You get what you pay for.
Cheers
Bob