In principle, this seems possible. The easiest way would probably be to put predictors on the unconstrained representation of the correlation matrix (see 10.9 Correlation matrices | Stan Reference Manual) and then apply the transformation yourself. @spinkney recently shared Stan implementation of the transform at Correlation matrix with positively constrained off-diagonals - #3 by spinkney although since you would not be putting priors on the transformed matrix (but presumably rather on the predictors), you must not add the log - Jacobian correction.
However, I would expect this to work quite badly in practice. In all models where I used correlations, a huge amount of data was necessary to learn the correlations with any precision. And you would need substantially more data to be able to decompose the correlations into several terms.
I also have no intuition whether putting predictors on the canonical partial correlations is a sensible thing to do from theoretical perspective (i.e. whether you would assume the CPCs to change monotonically with your predictors).