Backscatter communication (BSC) technology can enable ubiquitous deployment of low-cost sustainable wireless devices. In this work we investigate the efficacy of a full-duplex antenna array reader in overcoming the limited communication range bottleneck of monostatic BSCs. As performance is strongly influenced by the channel estimation (CE) quality, we first derive a novel least-squares estimator (LSE) for the forward and backward links between the reader and the tag, assuming that reciprocity holds. After defining the transceiver design at reader using this LSE, we optimize the energy allocation for the CE and information decoding phases, to maximize the average backscattered signal-to-noise ratio (SNR). The unimodality of this SNR in optimization variable along with a tight approximation for the global optimal design are also presented. Lastly, numerical results validate the proposed analysis and present key insights into the optimal LSE and energy allocation.