This paper addresses printed electronics as a promising alternative to silicon-based systems, requiring properties such as flexibility, stretchability, conformability, and ultra-low fabrication costs. Despite the large feature sizes of printed electronics, printed neural networks (NNs) have attracted significant attention for meeting target application requirements. However, implementing complex circuits remains challenging. This study addresses the gap between classification accuracy and area efficiency in printed neural networks by addressing the design and co-optimization of the entire processing-proximity-sensor system, from the analog-to-digital interface (a major area and power bottleneck) to the digital classifier. This study proposes an automated framework for designing printed ternary neural networks with arbitrary input precision, utilizing multi-objective optimization and global approximation. The proposed circuits are, on average, 17x more efficient in area and 59x more power efficient than conventional approximate printed neural networks, and are the first to enable printed battery-powered operation with less than a 5% accuracy loss while accounting for the cost of analog-to-digital interfacing.