If you want to model this in stan, then y is a random variable on equal footing with your other variables and you should define a probability distribution over it (the likelihood), usually conditioned on the remaining variables. I imagine your full joint distribution looks like
so the above transformation was more correctly for the conditional distribution p(y|\beta_0,\beta_1,x) and so does not require a Jacobian term adjustment for the betas.
I am not quite familiar with the model you are pursuing, but I think I get the gist.
In short, your original model specification is sufficient, but you might want to add priors to the betas. I think you can safely use priors designed for general regression problems.
beta0 ~ normal(0,1);
beta1 ~ normal(0,1);
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