More of a comment, but too long. This is a variant of *calibration*, or *inverse regression*/*inverse prediction*. One survey paper is [this at JSTOR](https://www.jstor.org/stable/1403690?seq=1#metadata_info_tab_contents) but it does not look explicitly at predicting the peak location.  One similar problem is response surface methods, and there is [a relevant R package](https://sites.psu.edu/engineeringstatistics/files/2016/09/OptimaRegionVignette-19nr0ts.pdf).  Some stored google searches that looks promosing: [peak detection and confidence intervals](https://www.google.com/search?q=peak+detection+and+confidence+intervals&oq=peak+detection+and+confidence+intervals&aqs=chrome..69i57.11448j0j7&sourceid=chrome&ie=UTF-8) and [response surface, inverse regression](https://www.google.com/search?q=calibration+%22inverse+regression%22+in+%22response+surface+methods%22&oq=calibration+%22inverse+regression%22+in+%22response+surface+methods%22&aqs=chrome..69i57.26380j0j7&sourceid=chrome&ie=UTF-8).  Among these I guess you can find something of interest!