We propose a minimal BER power allocation scheme for a class of quasi-orthogonal space-time block coded transmission (the so-called ABBA system) over i.i.d. Rayleigh fading channels under the QR-based successive detection framework. Instead of relying on BER under a fixed channel realization, the design criterion we adopt is the overall mean BER averaged with respect to the channel distribution. By exploiting a distinctive channel matrix structure induced by the ABBA code we derive a closed-form upper bound of the considered BER metric and, through minimization of which, an optimal power allocation scheme is obtained. Numerical simulation shows that the resultant performance is almost identical to that of the joint maximum-likelihood decoding in the medium-to-high SNR regime.