Properties of a custom wavelet family for continuous wavelet transform
It sounds like you don't need a wavelet. A wavelet's translation and dilation properties allow you to get both time and frequency resolution.

A matched filter takes an inner product of a signal with another signal, allowing you to detect a known signal in noise (by correlating the signal you have with the known signal). You may want to use this idea instead. 

Another idea is matching pursuit type algorithms -- if you have an overcomplete set of signals, and you want to find an approximation of a signal with respect to these overcomplete set of signals, you can do this through a matching pursuit algorithm. Basically, iteratively form a residual by finding the largest inner product signal in the set and then subtracting a scaled verison of the signal.