Is there any known CCC closed under a probabilistic powerdomain operation?
The following is an extended comment, it does not answer your question in the terms you posed it but does give a semantics for higher-order probabilistic calculi which you may find of interest.

In the past few years there has been a very active line of research around so-called quantitative denotational semantics of linear logic, based on the idea (originally due to Girard [1]) that higher-order programs may be modeled by power series.  In the probabilistic case, this takes the form of so-called probabilistic coherence spaces (PCS), also introduced by Girard [2] and studied in depth by Danos and Ehrhard [3].  PCS yield models of both typed and untyped probabilistic calculi which are of a very different nature than power domains and other monad-related models.  In particular, PCS give what is thus far the only known fully abstract model of probabilistic PCF [4], which is notoriously difficult and seemingly impossible to achieve with power domains (cf. the work of Jean Goubault-Larrecq).

Apart from Ehrhard, quantitative semantics is now actively developed by Michele Pagani and coauthors, I suggest you look at his web page for additional references.

[1] Jean-Yves Girard, Normal functors, power series and $\lambda$-calculus. Annals of Pure and Applied Logic 37 (2):129-177, 1988.

[2] Jean-Yves Girard, Between logic and quantic: a tract. In Linear logic in computer science, CUP, 2004.

[3] Vincent Danos and Thomas Ehrhard, Probabilistic coherence spaces as a model of higher-order probabilistic computation. Information and Computation 209(6): 966-991, 2011.

[4] Thomas Ehrhard, Michele Pagani and Christine Tasson, Probabilistic coherence spaces are fully abstract for probabilistic PCF. In Proceedings of POPL, pp. 309-320, 2014.