How to derive the stationary Kalman filter predictor?
Your derivations are correct.  

$\bar P = P(t|t-1)$ and $K(t) = A \bar K$

Is this your confusion: 


Why didn't they have the term $t|t-1$ in the Kalman Gain and Covariance Matrix Expressions?
How can this be "stationary" when your derivation shows that it is time varying? 





Bad choice of notation on the book's part


Let's look at the expression: 
$ \bar P = A\bar PA^T - A\bar P C^T(C\bar P C^T + R)^{-1}C\bar PA^T + Q $. 
The fact that $\bar P $ is a function of itself shows a recursive relationship. In other words, it uses its past values. So, it is NOT the same for all time instants -  It changes at every iteration. 


Misunderstanding of the word "stationary". 


When the author of the book said "stationary" he/she did not mean that $P$ and $K$ have the same value at all time instants. Instead, the author wanted to emphasise that the expressions for those to values are the same for all statistical realisations.  Stationarity is a statistical concept which means that the Statistics of the system is same all the time. Look at the expressions for $\bar P$ and $\bar K$ again. They depend only on 
\


Previous values of themselves
Transition matrices $A$ and $C$ which are deterministic and in your case time-invariant ($A$  and $C$ are the same at all times)
$Q$ and $R$ which are the noise covariance matrices. These 2 matrices describe the statistics of the noises and are the same in all realisations and time instances. 


The Kalman gain, $K$, and the state covariance Matrix $P$ will have the same value for all realisations of this random process. (Side Note: None of these 2 terms depend on the measurements, $y$. So they can be computed beforehand.)



Conclusion:

The "time-variant" equations you derived were equivalent to the ones in the book. Besides, the notational differences, there was a slight misunderstanding on your part regarding what changes and what doesn't.