If $\beth_1$ is weakly inaccessible, are any of the cardinal characteristics of continuum provably strictly less than $\beth_1$? Assume ZFC+"$\beth_1$ is weakly inaccessible." Are there any cardinal characteristics of the continuum mentioned at wikipedia that can thereby be proved to have cardinality strictly less than $\beth_1$?

No, assuming that theory is consistent. Martin's Axiom is known to imply that all those cardinal characteristics go to the top, with value continuum, but it is consistent with MA that the continuum is weakly inaccessible, since one may undertake the usual forcing of MA of length an inaccessible cardinal.
