If you've paid the lettings agencies (rip-off) referencing fee's then they should be able to use that information for any other property, I'd be asking them whether they have any suitable empty properties that you could switch to.
Asking for more deposit and rent, this close to the move in date is absurd, and if possible I'd want to cause the landlord some problems for being a ****
Asking for more deposit and rent, this close to the move in date is absurd, and if possible I'd want to cause the landlord some problems for being a ****