I am searching for a good category to think about probability theory in, with arrows as something like stochastic maps. There are certain nice structural features I would like the category to have (listed below) and I was hoping somebody could tell me if there are any categories meeting a lot of these conditions:

A monoidal category with a tensor product corresponding to something like the Cartesian product. Actually it would be good if it was a Markov category (like $\text{FinStoch}$ is for example).

In addition to having objects corresponding to finite sets/spaces it would be good to have infinite sets/spaces appearing as objects. Ideally I want to be able to talk about continuous probability distributions (using real numbers etc.) in addition to discrete probability distributions.

It would be great if I can form disjoint unions or coproducts of objects. Ideally I would like the category to be a bimonoidal category / rig category with $ \otimes $ corresponding to the Cartesian product and $\oplus$ corresponding to the disjoint union.

It would be great if the deterministic maps/functions from something like Set are included in the category too, so I can think about functional programming and recursion there too.

I realize I am asking for a lot, but there are many different categories for doing probability theory which are mentioned in the literature, so I was hoping somebody could highlight some categories which get close to fulfilling my conditions. In particular there seem to be many categories such as $\text{FinStoch}$, $\text{Stoch}$, $\text{CGStoch}$ and $\text{BorelStoch}$ defined by Fong and Fritz. Also 28:38 of Perrone's YouTube video defines more categories (like the category of finitary stochastic maps, which is the Kleisli category of Set using the distribution monad, and may meet some of my conditions).