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(Abridged) Recent XMM-Newton studies of X-ray variability in the hard states of black hole X-ray binaries (BHXRBs) imply that the variability is generated in the 'standard' optically-thick accretion disc. The variability originates in the disc as mass-accretion fluctuations and propagates through the disc to 'light up' inner disc regions, eventually modulating the power-law emission that is produced relatively centrally. We present a comparative spectral-timing study of XMM-Newton data from the BHXRB SWIFT J1753.5-0127 in a bright 2009 hard state with that from the significantly fainter 2006 hard state, to show for the first time the change in disc spectral-timing properties associated with a global increase in both the accretion rate and the relative contribution of the disc emission to the bolometric luminosity. We show that, although there is strong evidence for intrinsic disc variability in the more luminous hard state, the disc variability amplitude is suppressed relative to that of the power-law emission, which contrasts with the behaviour at lower luminosities where the disc variability is slightly enhanced when compared with the power-law variations. In the higher-luminosity data, the disc variability below 0.6 keV becomes incoherent with the power-law and higher-energy disc emission at frequencies below 0.5 Hz, in contrast with the coherent variations seen in the 2006 data. We explain these differences and the associated complex lags in the 2009 data in terms of the fluctuating disc model. If the variable signals are generated at small radii in the disc, the variability of disc emission can be naturally suppressed by the fraction of unmodulated disc emission from larger radii. The drop in coherence can be produced by disc accretion fluctuations arising at larger radii which are viscously damped and hence unable to propagate to the inner, power-law emitting region., 15 pages, 11 figures. Accepted for publication in MNRAS