You are asking about three choice principles, two of which have practically no research around them. Mainly due to the lack of tools we have for dealing with them.

The most you can find is the following paper,

> <cite authors="Howard, Paul; Tachtsis, Eleftherios">_Howard, Paul; Tachtsis, Eleftherios_, [**No decreasing sequence of cardinals**](http://dx.doi.org/10.1007/s00153-015-0472-5), Arch. Math. Logic 55, No. 3-4, 415-429 (2016). [ZBL1339.03038](https://zbmath.org/?q=an:1339.03038).</cite>

Where the authors show that $\sf AC_{WO}$ does not imply that there are no decreasing sequences of cardinals (which they take to mean actual sets, rather than the cardinals of these sets), at least in $\sf ZF$ (in $\sf ZFA$ we can say ever so slightly more).

This is really not a whole lot. As for $\sf BPI$, we know that it holds in Cohen's model, where there is a decreasing sequence of sets, simply because there are infinite Dedekind-infinite sets. But since we don't have any models of $\sf ZF+\lnot AC$ where we know that the cardinals are well-founded, we cannot prove it will not imply $\sf BPI$, mainly because it may very well imply $\sf AC$ for all we know.

So, you either need to come up with a completely new proof that somehow $\sf BPI$ follows from the well-foundedness of the cardinals, or prove that it implies choice outright, or better yet: develop new tools for positive results about cardinal structure in $\sf ZF$ that will let you deal with this sort of choice principle.