If the letting agents web site is listing trade organisations then maybe they are saying that any contractors they use are registered with the appropriate organisations.
As for the tiler not fitting an RCD, well as I see it no 1 he was not qualified to do so, 2 he removed a fitting and reinstalled the same fitting, so as far as I am aware even in a bathroom it is allowed with out being qualified at all.
As to the safety matter, well an RCD makes things safer but you have interfered with someone else's property without permission and then sent them a bill!!!
Your intentions are very good but I am not sure you did the right thing on a rented property.