When designing a system to transmit satellite images, it is important to balance the need for a high compression rate with the ability to accurately reconstruct them. The discrete wavelet transform (DWT) can be used to compress images and can be fine-tuned for specific classes of images, but it requires substantial hardware resources to perform floating-point multiplications. Separate research efforts to evolve wavelets and hardware multipliers have been performed, but not together. This work explores the use of several evolutionary and coevolutionary algorithms to address this gap, focusing on different ways of combining wavelets and hardware. The first algorithm evolves them in two separate stages, a second evolves them in pairs, and the third algorithm cooperatively coevolves two populations of wavelets and hardware. The results indicate that problem decomposition is most beneficial when evolution allows individual components to be optimized separately. Additionally, our evolved wavelets show improved accuracy over existing wavelets at high compression ratios, and the evolved hardware uses four times fewer components as a human-designed approach. This work demonstrates strong potential for effective image compression and, more broadly, the use of coevolutionary algorithms to fine-tune circuit design alongside secondary systems defining the needs of those circuits.