When I used the wavelength 240 I got over for reading by a microplate reader (Spectro) for concentrations 12,10, and 8 mg I dissolved in 2 ml DMSO and 4 ml ddH2O.
Not all microplate readers are capable of measuring such a short wavelength, but if you were able to set yours for 240 nm, then it must be capable of measuring it. The type of plate you use will be very important. You can't use polystyrene plates. You require either quartz or UV-transparent plastic plates.
If the absorbance measurements are too high despite using an appropriate plate, you can either dilute the sample or reduce the volume of the sample to lower the absorbance.
Light scattering (as distinct from absorbance) from nanoparticles will be very severe at 240 nm, so you will need a sample of unloaded NPs to use as a blank.