This paper presents printed electronics as a promising alternative to silicon-based systems for applications requiring flexibility, stretchability, conformability, and ultra-low fabrication costs. Despite the large size 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-sensor proximity system, from the analog-to-digital interface (a major area and power bottleneck) to the digital classifier. This paper proposes an automated framework for designing printed ternary neural networks with arbitrary input precision, utilizing multi-objective optimization and global approximation. The proposed circuit outperforms conventional approximate printed neural networks by an average of 17x in area and 59x in power, and is the first to enable printed battery-powered operation with less than 5% accuracy loss while considering the cost of analog-to-digital interfacing.