In  a method for removing the regularity conditions from the Cramer-Rao Inequality was given and applied to the estimation of a single real parameter. It was noted there that the method would extend to problems more general than estimating a single real parameter. However, the method extends also for the estimation of a single real parameter and produces analogues of the Bhattacharyya bounds with and without nuisance parameters.
"Bhattacharyya Bounds without Regularity Assumptions." Ann. Math. Statist. 23 (4) 629 - 632, December, 1952. https://doi.org/10.1214/aoms/1177729344