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 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. Some stored google searches that looks promosing: peak detection and confidence intervals and response surface, inverse regression. Among these I guess you can find something of interest!

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 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. Some stored google searches that looks promosing: peak detection and confidence intervals and response surface, inverse regression. In a comment whuber points to How to estimate the uncertainty in the zeros of a fitted function? which might be helpful.

Among these I guess you can find something of interest!