Since you are working with one-dimensional GPs you might be interested in the work that has gone into converting GPs with particular choices of kernel into Kalman filters.
The original work is by Simo Särkkä’s group, e.g.
J. Hartikainen and S. Särkkä, “Kalman filtering and smoothing solutions to temporal Gaussian process regression models,” in 2010 IEEE International Workshop on Machine Learning for Signal Processing, 2010, pp. 379–384.
but I also find this work from Oxford to be quite neat:
https://arxiv.org/abs/1510.02830