What you're proposing is to use time-varying effects or cofficients of covariates. To avoid confusion, restrict the phrase "time-varying covariates" to situations in which the values of covariates change over time. See the R time-varying vignette for the distinction.
Before you proceed to time-varying coefficients, try to improve the model to reduce the violation of proportional hazards (PH). Your model presently is restricted to a linear association of
p2 with log hazard, and only includes two predictors. A more flexible fit of
p2 and incorporation of other outcome-associated predictors might get around the PH violation. Alternatively, your data might better be handled with an accelerated-failure-time model that doesn't require PH (but comes with its own assumptions).
If you do end up needing to use time-varying coefficients in your model, there doesn't seem to be any easy computational test for PH thereafter. Although the R
survival package used to report results for a
cox.zph() test of PH on a model including the
tt() term used to handle time-varying coefficients, that calculation was incorrect and is no longer performed starting with version 3.1-2 of the
survival package. In the time-varying vignette linked above, the test for modeling PH properly thus is a visual comparison of shapes between the modeled time-dependence function and the smoothed estimates of variance-scaled Schoenfeld residuals over time from the model without the time-varying coefficient.
I'm not familiar with the
rstanarm* package, but in a quick look at its manual I didn't see that the package provides those residuals for evaluation. The plot you show from the
rstanarm documentation is only a comparison between the time-dependence of simulated data and what was returned by the function, so you can't really "show a plot like above for p1" for a model of empirical data. You might find a way to do a visual comparison like that in the
survival package documentation if you can find a way to extract the correct residuals from the
Nothing "automatically" assures that results will be "OK for publishing"; you need to be ready to document to the reader that your model is good enough for its purposes. If you choose to use a Bayesian survival model, make sure that you understand the principles very well and are prepared to defend the modeling to a skeptical reviewer. Advantages of Bayesian survival modeling have long been appreciated but still aren't what one typically sees in the literature, so your reviewers and readers might need more than the usual guidance to understand what you're doing.
*As I understand it, the Bayesian modeling used by
rstanarm isn't strictly a Cox model. Unlike a standard Cox model, it seems to fit the baseline hazard directly as part of the model.