[RossBencina]

This is more or less the approach that is taken by Dan Simon's "Optimal State Estimation" book that I came here to recommend: https://academic.csuohio.edu/simon-daniel/state-estimation/ All the prerequisites are covered prior to introducing the Kalman filter in chapter 5. Although Simon does not go through the information filter before introducing the Kalman filter, he discusses it later.

However, to understand recursive least squares, in particular the covariance matrix update you're going to need a firm grounding in probability and statistics. Simon makes the case that probability theory is a less strict pre-requisite than multiple-input-multiple-output (state space) linear systems theory (for which I can recommend Chen's "Linear System Theory and Design").

So I would argue that to understand Kalman filters you need to know state space systems modelling, both continuous time and discrete time discretisation methods (this provides the dynamics that describe the time-update step), plus you need to know enough multivariate statistics to understand how the Kalman filter propagates the gaussian random variables (i.e. the Kalman state) through the dynamics and back and forth through the measurement matrices.