You can use some standard Kalman Filter package in R for this - sspir, dlm, kfas, etc.
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Already this does not have an analytic solution, which is usually a prerequisite to deriving an update solution.
With a regularization constraint, it becomes a constrained optimization problem.
You can formulate similar model for logistic regression, $$y_t = logit(\beta_t\cdot x_t \varepsilon_t), \\ \beta_t = \beta_ \eta_t$$ as it will be non-linear, you will need to use non-linear filtering method from above packages - EKF or UKF. Side question: of the alternative calculations above, any comments on which is better from a precision standpoint?
References: chmike (2013) https://stats.stackexchange.com/a/79845/70282 Cook, John (n.d.) Accurately computing running variance Finch, Tony.
However, another approach that may be better suited to your needs is the use of the QR factorization with with low rank updates.