Predictive compression has always been considered an attractive solution for onboard compression thanks to its low computational demands and the ability to accurately control quality on a pixel-by-pixel basis. Traditionally, predictive compression focused on the lossless and near-lossless modes of operation where the maximum error can be bounded but the rate of the compressed image is variable. Fixed-rate is considered a challenging problem due to the dependencies between quantization and prediction in the feedback loop, and the lack of a signal representation that packs the signals energy into few coefficients as in the case of transform coding. In this paper, we show how it is possible to design a rate control algorithm suitable for onboard implementation by providing a general framework to select quantizers in each spatial and spectral region of the image and optimize the choice so that the desired rate is achieved with the best quality. In order to make the computational complexity suitable for onboard implementation, models are used to predict the rate-distortion characteristics of the prediction residuals in each image block. Such models are trained on-the-fly during the execution and small deviations in the output rate due to unmodeled behavior are automatically corrected as new data are acquired. The coupling of predictive coding and rate control allows the design of a single compression algorithm able to manage multiple encoding objectives. We tailor the proposed rate controller to the predictor defined by the CCSDS-123 lossless compression recommendation and study a new entropy coding stage based on the range coder in order to achieve an extension of the standard capable of managing all the following encoding objectives: lossless, variable-rate near-lossless (bounded maximum error), fixed-rate lossy (minimum average error), and any in-between case such as fixed-rate coding with a constraint on the maximum error. We show the performance of the proposed architecture on the CCSDS reference dataset for multispectral and hyperspectral image compression and compare it with state-of-the-art techniques based on transform coding such as the use of the CCSDS-122 Discrete Wavelet Transform encoder paired with the Pairwise Orthogonal Transform working in the spectral dimension. Remarkable results are observed by providing superior image quality both in terms of higher SNR and lower maximum error with respect to state-of-the-art transform coding.