We present a modified version of the two-arm, two-color, single second harmonic generation heterodyne dispersion interferometer, as introduced by Irby et. al. [Rev. Sci. Instrum.70, 699 (1999) 10.1063/1.1149489]. The amount of optical elements is reduced and digital in-phase and quadrature demodulation is used to retrieve the phase shift from a single photodetector signal. The intrinsic system noise and drift for this device are analyzed by measuring the Allan deviation. We investigate the use of this device for relative atmospheric pressure measurement. Relative pressure measurements are performed in a pressure chamber and referenced against a piezoresistive pressure transceiver to demonstrate the concept. It was found that the deviation was less than 150 Pa and an error estimation has been derived.