Observations of the Rossiter-McLaughlin (RM) effect provide information on star-planet alignments, which can inform planetary migration and evolution theories. Here, we go beyond the classical RM modelling and explore the impact of a convective blueshift that varies across the stellar disc and non-Gaussian stellar photospheric profiles. We simulated an aligned hot Jupiter with a 4 d orbit about a Sun-like star and injected centre-to-limb velocity (and profile shape) variations based on radiative 3D magnetohydrodynamic simulations of solar surface convection. The residuals between our modelling and classical RM modelling were dependent on the intrinsic profile width and v sin i; the amplitude of the residuals increased with increasing v sin i, and with decreasing intrinsic profile width. For slowly rotating stars the centre-to-limb convective variation dominated the residuals (with amplitudes of 10s of cm/s to ~1 m/s); however, for faster rotating stars the dominant residual signature was due a non-Gaussian intrinsic profile (with amplitudes from 0.5-9 m/s). When the impact factor was 0, neglecting to account for the convective centre-to-limb variation led to an uncertainty in the obliquity of ~10-20 degrees, even though the true v sin i was known. Additionally, neglecting to properly model an asymmetric intrinsic profile had a greater impact for more rapidly rotating stars (e.g. v sin i = 6 km/s), and caused systematic errors on the order of ~20 degrees in the measured obliquities. Hence, neglecting the impact of stellar surface convection may bias star-planet alignment measurements and consequently also theories on planetary migration and evolution.