I would write a letter, explaining all your concerns and the options that they have. either room by room or as a whole
good luck i hope the re-tile comes your way
Did you show the loose tiles to confirm your fears of the tiling job and that regrouting would be wrong in the first place?
Its all good and well being recommended by a friend but as most have said will be your reputation on the line,but getting them to do a full refurb should be done withthe original tiler who is at fault,they should get their money back and pay you to do the job properly a professional tiler
I would tell them you won't regrout and the reasons why. Tell them it needs a rip out and that if they want a qoute to get in touch. By not quoting there and then your giving the breathing space and removing any doubt your just trying to grab there cash .